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University Physics with Modern Physics

Hugh D. Young

Chapter 2

Motion Along a Straight Line - all with Video Answers

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Chapter Questions

00:57

Problem 1

A car travels in the $+x$ -direction on a straight and level road. For the first 4.00 s of its motion, the average velocity of the car is $v_{\mathrm{av}-x}=6.25 \mathrm{m} / \mathrm{s} .$ How far does the car travel in 4.00 $\mathrm{s}$ ?

Zachary Warner
Zachary Warner
Numerade Educator
03:04

Problem 2

In an experiment, a shearwater (a seabird) was taken from its nest, flown 5150 $\mathrm{km}$ away, and released. The bird found its way back to its nest 13.5 days after release. If we place the origin in the nest and extend the $+x$ -axis to the release point, what was the bird's average velocity in $\mathrm{m} / \mathrm{s}($ a) for the return flight, and (b) for the whole episode, from leaving the nest to returning?

Zachary Warner
Zachary Warner
Numerade Educator
02:28

Problem 3

Trip Home. You normally drive on the freeway between San Diego and Los Angeles at an average speed of 105 $\mathrm{km} / \mathrm{h}$ (65 $\mathrm{mi} / \mathrm{h} ),$ and the trip takes 2 $\mathrm{h}$ and 20 min. On a Friday afternoon, however, heavy traffic slows you down and you drive the same distance at an average speed of only 70 $\mathrm{km} / \mathrm{h}(43 \mathrm{mi} / \mathrm{h})$ How much longer does the trip take?

Zachary Warner
Zachary Warner
Numerade Educator
04:35

Problem 4

From Pillar to Post. Starting from a pillar, you run 200 $\mathrm{m}$ east (the $+x$ -direction) at an average speed of $5.0 \mathrm{m} / \mathrm{s},$ and then run 280 $\mathrm{m}$ west at an average speed of 4.0 $\mathrm{m} / \mathrm{s}$ to a post. Calculate (a) your average speed from pillar to post and (b) your average velocity from pillar to post.

Zachary Warner
Zachary Warner
Numerade Educator
02:30

Problem 5

Starting from the front door of your ranch house, you walk 60.0 $\mathrm{m}$ due east to your windmill, and then you turn around and slowly walk 40.0 $\mathrm{m}$ west to a bench where you sit and watch the sunrise. It takes you 28.0 s to walk from your house to the windmill and then 36.0 s to walk from the windmill to the bench. For the entire trip from your front door to the bench, what are (a) your average velocity and (b) your average speed?

Zachary Warner
Zachary Warner
Numerade Educator
03:29

Problem 6

A Honda Civic travels in a straight line along a road. Its distance $x$ from a stop sign is given as a function of time $t$ by the equation $x(t)=\alpha t^{2}-\beta t^{3},$ where $\alpha=1.50 \mathrm{m} / \mathrm{s}^{2}$ and $\beta=$ 0.0500 $\mathrm{m} / \mathrm{s}^{3} .$ Calculate the average velocity of the car for each time interval: $(\mathrm{a}) t=0$ to $t=2.00 \mathrm{s} ;$ (b) $t=0$ to $t=4.00 \mathrm{s}$ ; (c) $t=2.00$ s to $t=4.00 \mathrm{s}.$

Zachary Warner
Zachary Warner
Numerade Educator
04:32

Problem 7

CALC A car is stopped at a traffic light. It then travels along a straight road so that its distance from the light is given by $x(t)=b t^{2}-c t^{3},$ where $b=2.40 \mathrm{m} / \mathrm{s}^{2}$ and $c=0.120 \mathrm{m} / \mathrm{s}^{3} .$ (a) Calculate the average velocity of the car for the time interval $t=0$ to $t=10.0$ s. (b) Calculate the instantaneous velocity of the car at $t=0, t=5.0 \mathrm{s},$ and $t=10.0 \mathrm{s}$ . (c) How long after starting from rest is the car again at rest?

Zachary Warner
Zachary Warner
Numerade Educator
02:03

Problem 8

A bird is flying due east. Its distance from a tall building is given by $x(t)=28.0 \mathrm{m}+(12.4 \mathrm{m} / \mathrm{s}) t-\left(0.0450 \mathrm{m} / \mathrm{s}^{3}\right) t^{3}$ What is the instantaneous velocity of the bird when $t=8.00 \mathrm{s}?$

Zachary Warner
Zachary Warner
Numerade Educator
05:03

Problem 9

A ball moves in a straight line (the $x$ -axis). The graph in Fig. E2.9 shows this ball's velocity as a function of time. (a) What are the ball's average speed and average velocity during the first 3.0 s? (b) Suppose that the ball moved in such a way that the graph segment after 2.0 s was $-3.0 \mathrm{m} / \mathrm{s}$ instead of $+3.0 \mathrm{m} / \mathrm{s} .$ Find the ball's average speed and average velocity in this case.

João Gabriel Alencar Caribé
João Gabriel Alencar Caribé
Numerade Educator
03:32

Problem 10

A physics professor leaves her house and walks along the sidewalk toward campus. After 5 min it starts to rain and she returns home. Her distance from her house as a function of time is shown in Fig. E2. $10 .$ At which of the labeled points is her velocity (a) zero? (b) constant and positive? (c) constant and negative? (d) increasing in magnitude? (e) decreasing in magnitude?

Zachary Warner
Zachary Warner
Numerade Educator
04:42

Problem 11

A test car travels in a straight line along the $x$-axis. The graph in Fig. E2. 11 shows the car's position $x$ as a function of time. Find its instantaneous velocity at points $A$ through $G.$

Zachary Warner
Zachary Warner
Numerade Educator
06:32

Problem 12

Figure E2.12 shows the velocity of a solar-powered car as a function of time. The driver accelerates from a stop sign, cruises for 20 s at a constant speed of 60 $\mathrm{km} / \mathrm{h}$ , and then brakes to come to a stop 40 $\mathrm{s}$ after leaving the stop sign. (a) Compute the average acceleration during the following time intervals: (i) $t=0$ to $t=10 \mathrm{s} ;(\mathrm{ii}) t=30 \mathrm{s}$ to $t=40 \mathrm{s} ;(\mathrm{iii}) t=10 \mathrm{s}$ to $t=30 \mathrm{s} ;$ (iv) $t=0$ to $t=40 \mathrm{s}$ . (b) What is the instantaneous acceleration at $t=20 \mathrm{s}$ and at $t=35 \mathrm{s} ?$

Zachary Warner
Zachary Warner
Numerade Educator
05:35

Problem 13

The Fastest (and Most Expensive) Car! The table shows test data for the Bugatti Veyron, the fastest car made. The car is moving in a straight line (the $x$-axis).

$\begin{array}{llll}{\text { Time }(s)} & {0} & {2.1} & {20.0} & {53} \\ {\text { Speed (mijh) }} & {0} & {60} & {200} & {253}\end{array}$

(a) Make a $v_{x}-t$ graph of this car's velocity $($ in $\mathrm{mi} / \mathrm{h})$ as a function of time. Is its acceleration constant? (b) Calculate the car's average acceleration ( in $\mathrm{m} / \mathrm{s}^{2} )$ between ( i 0 and $2.1 \mathrm{s} ;$ (ii) 2.1 $\mathrm{s}$ and 20.0 $\mathrm{s}$; (iii) 20.0 s and 53 s. Are these results consistent with your graph in part (a)? (Before you decide to buy this car, it might be helpful to know that only 300 will be built, it runs out of gas in 12 minutes at top speed, and it costs $\$ 1.25$ million!)

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:22

Problem 14

A race car starts from rest and travels east along a straight and level track. For the first 5.0 s of the car's motion, the eastward component of the car's velocity is given by $v_{x}(t)=\left(0.860 \mathrm{m} / \mathrm{s}^{3}\right) t^{2} .$ What is the acceleration of the car when $v_{x}=16.0 \mathrm{m} / \mathrm{s} ?$

Zachary Warner
Zachary Warner
Numerade Educator
17:36

Problem 15

A turtle crawls along a straight line, which we will call the $x$ -axis with the positive direction to the right. The equation for the turtle's position as a function of time is $x(t)=50.0 \mathrm{cm}+$ $(2.00 \mathrm{cm} / \mathrm{s}) t-\left(0.0625 \mathrm{cm} / \mathrm{s}^{2}\right) t^{2}$ (a) Find the turtle's initial velocity, initial position, and initial acceleration. (b) At what time $t$ is the velocity of the turtle zero? (c) How long after starting does it take the turtle to return to its starting point? (d) At what times $t$ is the turtle a distance of 10.0 $\mathrm{cm}$ from its starting point? What is the velocity (magnitude and direction) of the turtle at each of these times? (e) Sketch graphs of $x$ versus $t, v_{x}$ versus $t,$ and $a_{x}$ versus $t$ for the time interval $t=0$ to $t=40 \mathrm{s}.$

Zachary Warner
Zachary Warner
Numerade Educator
03:52

Problem 16

An astronaut has left the International Space Station to test a new space scooter. Her partner measures the following velocity changes, each taking place in a 10 -s interval. What are the magnitude, the algebraic sign, and the direction of the average acceleration in each interval? Assume that the position is to the right. (a) At the beginning of the interval the astronaut is moving toward the right along the $x$ -axis at $15.0 \mathrm{m} / \mathrm{s},$ and at the end of the interval she is moving toward the right at 5.0 $\mathrm{m} / \mathrm{s} .$ (b) At the beginning she is moving toward the left at $5.0 \mathrm{m} / \mathrm{s},$ and at the end she is moving toward the left at 15.0 $\mathrm{m} / \mathrm{s}$ (c) At the beginning she is moving toward the right at $15.0 \mathrm{m} / \mathrm{s},$ and at the end she is moving toward the left at 15.0 $\mathrm{m} / \mathrm{s}$.

Zachary Warner
Zachary Warner
Numerade Educator
05:02

Problem 17

A car's velocity as a function of time is given by $v_{x}(t)=\alpha+\beta t^{2},$ where $\alpha=3.00 \mathrm{m} / \mathrm{s}$ and $\beta=0.100 \mathrm{m} / \mathrm{s}^{3} .$ (a) Calculate the average acceleration for the time interval $t=0$ to $t=5.00$ s. (b) Calculate the instantaneous acceleration for $t=0$ and $t=5.00$ s. (c) Draw $v_{x^{-}} t$ and $a_{x^{-}}$ graphs for the car's motion between $t=0$ and $t=5.00 \mathrm{s}.$

Zachary Warner
Zachary Warner
Numerade Educator
08:43

Problem 18

The position of the front bumper of a test car under microprocessor control is given by $x(t)=2.17 \mathrm{m}+$ $\left(4.80 \mathrm{m} / \mathrm{s}^{2}\right) t^{2}-\left(0.100 \mathrm{m} / \mathrm{s}^{6}\right) t^{6} .$ (a) Find its position and acceleration at the instants when the car has zero velocity. (b) Draw $x$ -t, $v_{x}-t,$ and $a_{x}-t$ graphs for the motion of the bumper between $t=0$ and $t=2.00 \mathrm{s}.$

Zachary Warner
Zachary Warner
Numerade Educator
02:20

Problem 19

An antelope moving with constant acceleration covers the distance between two points 70.0 $\mathrm{m}$ apart in 7.00 $\mathrm{s} .$ Its speed as it passes the second point is 15.0 $\mathrm{m} / \mathrm{s}$ . (a) What is its speed at the first point? (b) What is its acceleration?

Zachary Warner
Zachary Warner
Numerade Educator
03:33

Problem 20

A jet fighter pilot wishes to accelerate from rest at a constant acceleration of 5$g$ to reach Mach 3 (three times the speed of sound) as quickly as possible. Experimental tests reveal that he will black out if this acceleration lasts for more than 5.0 s. Use 331 $\mathrm{m} / \mathrm{s}$ for the speed of sound. (a) Will the period of acceleration last long enough to cause him to black out? (b) What is the greatest speed he can reach with an acceleration of 5$g$ before blacking out?

Zachary Warner
Zachary Warner
Numerade Educator
02:15

Problem 21

The fastest measured pitched baseball left the pitcher's hand at a speed of 45.0 $\mathrm{m} / \mathrm{s}$ . If the pitcher was in contact with the ball over a distance of 1.50 $\mathrm{m}$ and produced constant acceleration, (a) what acceleration did he give the ball, and (b) how much time did it take him to pitch it?

Zachary Warner
Zachary Warner
Numerade Educator
02:13

Problem 22

A Tennis Serve. In the fastest measured tennis serve, the ball left the racquet at 73.14 $\mathrm{m} / \mathrm{s} .$ A served tennis ball is typically in contact with the racquet for 30.0 $\mathrm{ms}$ and starts from rest. Assume constant acceleration. (a) What was the ball's acceleration during this serve? (b) How far did the ball travel during the serve?

Zachary Warner
Zachary Warner
Numerade Educator
01:50

Problem 23

The human body can survive an acceleration trauma incident (sudden stop) if the magnitude of the acceleration is less than 250 $\mathrm{m} / \mathrm{s}^{2} .$ If you are in an automobile accident with an initial speed of 105 $\mathrm{km} / \mathrm{h}$ (65 mi/h) and you are stopped by an airbag that inflates from the dashboard, over what distance must the airbag stop you for you to survive the crash?

Zachary Warner
Zachary Warner
Numerade Educator
02:20

Problem 24

If a pilot accelerates at more than $4 g,$ he begins to "gray out" but doesn't completely lose consciousness. (a) Assuming constant acceleration, what is the shortest time that a jet pilot starting from rest can take to reach Mach 4 (four times the speed of sound) without graying out? (b) How far would the plane travel during this period of acceleration? (Use 331 $\mathrm{m} / \mathrm{s}$ for the speed of sound in cold air.)

Zachary Warner
Zachary Warner
Numerade Educator
02:19

Problem 25

Air-Bag Injuries. During an auto accident, the vehicle's air bags deploy and slow down the passengers more gently than if they had hit the windshield or steering wheel. According to safety standards, the bags produce a maximum acceleration of 60$g$ that lasts for only 36 $\mathrm{ms}$ (or less). How far (in meters) does a person travel in coming to a complete stop in 36 $\mathrm{ms}$ at a constant acceleration of 60$g?$

Zachary Warner
Zachary Warner
Numerade Educator
02:31

Problem 26

Prevention of Hip Fractures. Falls resulting in hip fractures are a major cause of injury and even death to the elderly. Typically, the hip's speed at impact is about 2.0 $\mathrm{m} / \mathrm{s} .$ If this can be reduced to 1.3 $\mathrm{m} / \mathrm{s}$ or less, the hip will usually not fracture. One way to do this is by wearing elastic hip pads. (a) If a typical pad is 5.0 $\mathrm{cm}$ thick and compresses by 2.0 $\mathrm{cm}$ during the impact of a fall, what constant acceleration $\operatorname{m} / \mathrm{s}^{2}$ and in $g^{\prime} \mathrm{s}$ ) does the hip undergo to reduce its speed from 2.0 $\mathrm{m} / \mathrm{s}$ to 1.3 $\mathrm{m} / \mathrm{s} ?$ (b) The acceleration you found in part (a) may seem rather large, but to fully assess its effects on the hip, calculate how long it lasts.

Zachary Warner
Zachary Warner
Numerade Educator
03:41

Problem 27

Are We Martians? It has been suggested, and not facetiously, that life might have originated on Mars and been carried to the earth when a meteor hit Mars and blasted pieces of rock (perhaps containing primitive life) free of the surface. Astronomers know that many Martian rocks have come to the earth this way. (For information on one of these, search the Internet for "ALH $84001 . "$ ') One objection to this idea is that microbes would have to undergo an enormous lethal acceleration during the impact. Let us investigate how large such an acceleration might be. To escape Mars, rock fragments would have to reach its escape velocity of $5.0 \mathrm{km} / \mathrm{s},$ and this would most likely happen over a distance of about 4.0 $\mathrm{m}$ during the meteor impact. (a) What would be the acceleration ( in $\mathrm{m} / \mathrm{s}^{2}$ and $g^{\prime} \mathrm{s}$) of such a rock fragment, if the acceleration is constant? (b) How long would this acceleration last? (c)
In tests, scientists have found that over 40$\%$ of Bacillius subtilis bacteria survived after an acceleration of $450,000 g .$ In light of your answer to part (a), can we rule out the hypothesis that life might have been blasted from Mars to the earth?

Zachary Warner
Zachary Warner
Numerade Educator
02:10

Problem 28

Entering the Freeway. A car sits in an entrance ramp to a freeway, waiting for a break in the traffic. The driver accelerates with constant acceleration along the ramp and onto the freeway. The car starts from rest, moves in a straight line, and has a speed of 20 $\mathrm{m} / \mathrm{s}(45 \mathrm{mi} / \mathrm{h})$ when it reaches the end of the $120-\mathrm{m}$ -long ramp. (a) What is the acceleration of the car? (b) How much time does it take the car to travel the length of the ramp? (c) The traffic on the freeway is moving at a constant speed of 20 $\mathrm{m} / \mathrm{s} .$ What distance does the traffic travel while the car is moving the length of the ramp?

Zachary Warner
Zachary Warner
Numerade Educator
03:54

Problem 29

Launch of the Space Shuttle. At launch the space shuttle weighs 4.5 million pounds. When it is launched from rest, it takes 8.00 s to reach 161 $\mathrm{km} / \mathrm{h}$ , and at the end of the first 1.00 $\mathrm{min}$ its speed is 1610 $\mathrm{km} / \mathrm{h}$ . (a) What is the average acceleration ($\operatorname{in}$ $\mathrm{m} / \mathrm{s}^{2}$ ) of the shuttle (i) during the first $8.00 \mathrm{s},$ and (ii) between 8.00 $\mathrm{s}$ and the end of the first 1.00 $\mathrm{min}$ ? (b) Assuming the acceleration is constant during each time interval (but not necessarily the same in both intervals, what distance does the shuttle travel (i) during the first $8.00 \mathrm{s},$ and ( ii) during the interval from 8.00 s to 1.00 $\mathrm{min}$?

Zachary Warner
Zachary Warner
Numerade Educator
08:18

Problem 30

A cat walks in a straight line, which we shall call the $x$ -axis with the positive direction to the right. As an observant physicist, you make measurements of this cat's motion and con- struct a graph of the feline's velocity as a function of time (Fig. $E 2.30$ ) (a) Find the cat's velocity at $t=4.0$ s and at $t=7.0 \mathrm{s} .$ (b) What is the cat's acceleration at $t=3.0 \mathrm{s} ?$ At $t=6.0 \mathrm{s} ?$ At $t=7.0 \mathrm{s} ?(\mathrm{c})$ What distance does the cat move during the first 4.5 s? From $t=0$ to $t=7.5 \mathrm{s} ?$ (d) Sketch clear graphs of the cat's acceleration and position as functions of time, assuming that the cat started at the origin.

Zachary Warner
Zachary Warner
Numerade Educator
07:31

Problem 31

The graph in Fig. E2.31 shows the velocity of a motorcycle police officer plotted as a function of time. (a) Find the instantaneous acceleration at $t=3$ s, at $t=7$ s, and at $t=11$ s. (b) How far does the officer go in the first 5 s? The first 9 s? The first 13 s?.

Zachary Warner
Zachary Warner
Numerade Educator
08:03

Problem 32

Two cars, $A$ and $B$ , move along the $x$ -axis. Figure E2.32 is a graph of the positions of $A$ and $B$ versus time. (a) In motion diagrams (like Figs. 2.13 $\mathrm{b}$ and grams flike Figs. 2.13 $\mathrm{b}$ and 2.14 $\mathrm{b}$ ), show the position, velocity, and acceleration of each of the two cars at $t=0, t=1 \mathrm{s}$ and $t=3$ s. (b) At what time $(\mathrm{s})$ if any, do $A$ and $B$ have the same position? (c) Graph velocity versus time for both $A$ and $B .(\mathrm{d})$ At what time $(\mathrm{s}),$ if any, do $A$ and $B$ have the same velocity? (e) At what time(s), if any, does car $A$ pass car $B ?(\mathrm{f})$ At what time $(\mathrm{s}),$ if any, does car $B$ pass car $A$ ?

Zachary Warner
Zachary Warner
Numerade Educator
07:55

Problem 33

Mars Landing. In January $2004,$ NASA landed exploration vehicles on Mars. Part of the descent consisted of the following stages:

Stage $A:$ Friction with the atmosphere reduced the speed from $19,300 \mathrm{km} / \mathrm{h}$ to 1600 $\mathrm{km} / \mathrm{h}$ in 4.0 $\mathrm{min.}$
Stage $B:$ A parachute then opened to slow it down to 321 $\mathrm{km} / \mathrm{h}$ in 94 $\mathrm{s} .$
Stage $C :$ Retro rockets then fired to reduce its speed to zero over a distance of 75 $\mathrm{m}.$

Assume that each stage followed immediately after the preceding one and that the acceleration during each stage was constant.
(a) Find the rocket's acceleration (in $\mathrm{m} / \mathrm{s}^{2}$ ) during each stage.
(b) What total distance (in km) did the rocket travel during stages $\mathrm{A}, \mathrm{B},$ and $\mathrm{C} ?$

Zachary Warner
Zachary Warner
Numerade Educator
08:40

Problem 34

At the instant the traffic light turns green, a car that has been waiting at an intersection starts ahead with a constant acceleration of 3.20 $\mathrm{m} / \mathrm{s}^{2} .$ At the same same instant a truck, traveling with a constant speed of $20.0 \mathrm{m} / \mathrm{s},$ overtakes and passes the car. (a) How far beyond its starting point does the car overtake the truck? (b) How fast is the car traveling when it overtakes the truck? (c) Sketch an $x-t$ graph of the motion of both vehicles. Take $x=0$ at the intersection. (d) Sketch a $v_{x}-t$ graph of the motion of both vehicles.

Zachary Warner
Zachary Warner
Numerade Educator
02:51

Problem 35

(a) If a flea can jump straight up to a height of $0.440 \mathrm{m},$ what is its initial speed as it leaves the ground? (b) How long is it in the air?

Zachary Warner
Zachary Warner
Numerade Educator
05:57

Problem 36

A small rock is thrown vertically upward with a speed of 18.0 $\mathrm{m} / \mathrm{s}$ from the edge of the roof of a 30.0 -m-tall building. The rock doesn't hit the building on its way back down and lands in the street below. Air resistance can be neglected. (a) What is the speed of the rock just before it hits the street? (b) How much time elapses from when the rock is thrown until it hits the street?

Dading Chen
Dading Chen
Numerade Educator
01:44

Problem 37

A juggler throws a bowling pin straight up with an initial speed of 8.20 $\mathrm{m} / \mathrm{s} .$ How much time elapses until the bowling pin returns to the juggler's hand?

Zachary Warner
Zachary Warner
Numerade Educator
02:23

Problem 38

You throw a glob of putty straight up toward the ceiling, which is 3.60 $\mathrm{m}$ above the point where the putty leaves your hand. The initial speed of the putty as it leaves your hand is 9.50 $\mathrm{m} / \mathrm{s}$. (a) What is the speed of the putty just before it strikes the ceiling?
(b) How much time from when it leaves your hand does it take the putty to reach the ceiling?

Zachary Warner
Zachary Warner
Numerade Educator
07:42

Problem 39

A tennis ball on Mars, where the acceleration due to gravity is 0.379$g$ and air resistance is negligible, is hit directly upward and returns to the same level 8.5 s later. (a) How high above its original point did the ball go? (b) How fast was it moving just after being hit? (c) Sketch graphs of the ball's vertical position, vertical velocity, and vertical acceleration as functions of time while it's in the Martian air.

Zachary Warner
Zachary Warner
Numerade Educator
01:12

Problem 40

Touchdown on the Moon. A lunar lander is making its descent to Moon Base I (Fig. E2.40). The lander descends slowly under the retrothrust of its descent engine. The engine is cut off when the lander is 5.0 $\mathrm{m}$ above the surface and has a downward speed of 0.8 $\mathrm{m} / \mathrm{s} .$ With the engine off, the lander is in free fall. What is the speed of the lander just before it touches the surface? The acceleration due to gravity on the moon is 1.6 $\mathrm{m} / \mathrm{s}^{2}$.

Dading Chen
Dading Chen
Numerade Educator
02:51

Problem 41

A Simple Reaction-Time Test. A meter stick is held vertically above your hand, with the lower end between your thumb and first finger. On seeing the meter stick released, you grab it with these two fingers. You can calculate your reaction time from the distance the meter stick falls, read directly from the point where your fingers grabbed it. (a) Derive a relationship for your reaction time in terms of this measured distance, $d$ (b) If the measured distance is $17.6 \mathrm{cm},$ what is the reaction time?

Zachary Warner
Zachary Warner
Numerade Educator
05:08

Problem 42

A brick is dropped (zero initial speed) from the roof of a building. The brick strikes the ground in 2.50 s. You may ignore air resistance, so the brick is in free fall. (a) How tall, in meters, is the building? (b) What is the magnitude of the brick's velocity just before it reaches the ground? (c) Sketch $a_{y}-t, v_{y}-t,$ and $y-t$ graphs for the motion of the brick.

Zachary Warner
Zachary Warner
Numerade Educator
12:33

Problem 43

Launch Failure. $A 7500$ -kg rocket blasts off vertically from the launch pad with a constant upward acceleration of 2.25 $\mathrm{m} / \mathrm{s}^{2}$ and feels no appreciable air resistance. When it has reached a height of $525 \mathrm{m},$ its engines suddenly fail so that the only force acting on it is now gravity. (a) What is the maximum height this rocket will reach above the launch pad? (b) How much time after engine failure will elapse before the rocket comes crashing down to the launch pad, and how fast will it be moving just before it crashes? (c) Sketch $a_{y^{-}} t, v_{y^{-}} t,$ and $y-t$ graphs of the rocket's motion from the instant of blast-off to the instant just before it strikes the launch pad.

Zachary Warner
Zachary Warner
Numerade Educator
11:15

Problem 44

A hot-air balloonist, rising vertically with a constant velocity of magnitude 5.00 $\mathrm{m} / \mathrm{s}$ releases a sandbag at an instant when the balloon is 40.0 $\mathrm{m}$ above the ground (Fig. E2.44). After it is released, the sandbag is in free fall. (a) Compute the position and velocity of the sandbag at 0.250 s and 1.00 after its release. (b) How many seconds after its release will the bag strike the ground? (c) With what magnitude of velocity does it strike the ground? (d) What is the greatest height above the ground that the sandbag reaches? (e) Sketch $a_{y}-t, v_{y}-t,$ and $y-t$ graphs for the motion.

Zachary Warner
Zachary Warner
Numerade Educator
04:14

Problem 45

The rocket-driven sled Sonic Wind No. $2,$ used for investigating the physiological effects of large accelerations, runs on a straight, level track 1070 $\mathrm{m}(3500 \mathrm{ft})$ long. Starting from rest, it can reach a speed of 224 $\mathrm{m} / \mathrm{s}(500 \mathrm{mi} / \mathrm{h})$ in 0.900 $\mathrm{s.}$ (a) Com- pute the acceleration in $\mathrm{m} / \mathrm{s}^{2}$ , assuming that it is constant. (b) What is the ratio of this acceleration to that of a freely falling body $(g) ?(\mathrm{c})$ (c) What distance is covered in 0.900 $\mathrm{s} ?$ (d) A magazine article states that at the end of a certain run, the speed of the sled decreased from 283 $\mathrm{m} / \mathrm{s}(632 \mathrm{mi} / \mathrm{h})$ to zero in 1.40 $\mathrm{s}$ and that during this time the magnitude of the acceleration was greater than 40$g .$ Are these figures consistent?

Zachary Warner
Zachary Warner
Numerade Educator
05:43

Problem 46

An egg is thrown nearly vertically upward from a point near the cornice of a tall building. It just misses the cornice on the way down and passes a point 30.0 m below its starting point 5.00 $\mathrm{s}$ after it leaves the thrower's hand. Air resistance may be ignored. (a) What is the initial speed of the egg? (b) How high does it rise above its starting point? (c) What is the magnitude of its velocity at the highest point? (d) What are the magnitude and direction of its acceleration at the highest point? (e) Sketch $a_{y}-t, v_{y}-t,$ and $y-t$ graphs for the motion of the egg.

Zachary Warner
Zachary Warner
Numerade Educator
03:55

Problem 47

A 15-kg rock is dropped from rest on the earth and reaches the ground in 1.75 $\mathrm{s}.$ When it is dropped from the same height on Saturn's satellite Enceladus, it reaches the ground in 18.6 $\mathrm{s}.$ What is the acceleration due to gravity on Enceladus?

Zachary Warner
Zachary Warner
Numerade Educator
07:23

Problem 48

A large boulder is ejected vertically upward from a volcano with an initial speed of 40.0 $\mathrm{m} / \mathrm{s} .$ Air resistance may be ignored. (a) At what time after being ejected is the boulder moving at 20.0 $\mathrm{m} / \mathrm{s}$ upward? (b) At what time is it moving at 20.0 $\mathrm{m} / \mathrm{s}$ downward? (c) When is the displacement of the boulder from its initial position zero? (d) When is the velocity of the boulder zero? (e) What are the magnitude and direction of the acceleration while the boulder is (i) moving upward? (ii) Moving downward? (iii) At the highest point? (f) Sketch $a_{y}-t, v_{y}-t,$ and $y-$ t graphs for the motion.

Zachary Warner
Zachary Warner
Numerade Educator
05:50

Problem 49

Two stones are thrown vertically upward from the ground, one with three times the initial speed of the other. (a) If the faster stone takes 10 s to return to the ground, how long will it take the slower stone to return? (b) If the slower stone reaches a maximum height of $H$ , how high (in terms of $H )$ will the faster stone go? Assume free fall.

Zachary Warner
Zachary Warner
Numerade Educator
03:34

Problem 50

For constant $a_{x},$ use Eqs. $(2.17)$ and $(2.18)$ to find $v_{x}.$ and $x$ as functions of time. Compare your results to Eqs. $(2.8)$ and $(2.12).$

Zachary Warner
Zachary Warner
Numerade Educator
07:10

Problem 51

A rocket starts from rest and moves upward from the surface of the earth. For the first 10.0 s of its motion, the vertical acceleration of the rocket is given by $a_{y}=\left(2.80 \mathrm{m} / \mathrm{s}^{3}\right) t,$ where acceleration of the rocket is given by $a_{y}=\left(2.80 \mathrm{m} / \mathrm{s}^{3}\right) t,$ where the $+y$ -direction is upward. (a) What is the height of the rocket above the surface of the earth at $t=10.0 \mathrm{s} ?$ (b) What is the speed of the rocket when it is 325 $\mathrm{m}$ above the surface of the earth?

Zachary Warner
Zachary Warner
Numerade Educator
04:48

Problem 52

The acceleration of a bus is given by $a_{x}(t)=\alpha t$ , where $\alpha=1.2 \mathrm{m} / \mathrm{s}^{3} .$ (a) If the bus's velocity at time $t=1.0 \mathrm{s}$ is 5.0 $\mathrm{m} / \mathrm{s},$ what is its velocity at time $t=2.0 \mathrm{s} ?$ (b) If the bus's position at time $t=1.0 \mathrm{s}$ is $6.0 \mathrm{m},$ what is its position at time $t=2.0 \mathrm{s} ?(\mathrm{c}) \operatorname{Sketch} a_{x}-t, v_{x}-t,$ and $x$ -t graphs for the motion.

Zachary Warner
Zachary Warner
Numerade Educator
06:10

Problem 53

The acceleration of a motorcycle is given by $a_{x}(t)=A t-B t^{2},$ where $A=1.50 \mathrm{m} / \mathrm{s}^{3}$ and $B=0.120 \mathrm{m} / \mathrm{s}^{4}$. The motorcycle is at rest at the origin at time $t=0 .$ (a) Find its position and velocity as functions of time. (b) Calculate the maximum velocity it attains.

Zachary Warner
Zachary Warner
Numerade Educator
08:13

Problem 54

Flying Leap of the Flea. High-speed motion pictures $(3500$ frames/second) of a jumping, $210-\mu \mathrm{g}$ flea yielded the data used to plot the graph given in Fig. E2.54. (See "The Flying Leap of the Flea" by M. Rothschild, Y. Schlein, K. Parker, C. Neville, and S. Sternberg in the November 1973 Scientific American.) This flea was about 2 $\mathrm{mm}$ long and jumped at a nearly vertical takeoff angle. Use the graph to answer the questions. (a) Is the acceleration of the flea ever zero? If so, when? Justify your answer. (b) Find the maximum height the flea reached in the first 2.5 ms. (c) Find the flea's acceleration at $0.5 \mathrm{ms}, 1.0 \mathrm{ms},$ and 1.5 $\mathrm{ms}$ . (d) Find the flea's height at $0.5 \mathrm{ms}, 1.0 \mathrm{ms},$ and 1.5 $\mathrm{ms}$ .

Zachary Warner
Zachary Warner
Numerade Educator
06:30

Problem 55

A typical male sprinter can maintain his maximum acceleration for $2.0 \mathrm{~s}$ and his maximum speed is $10 \mathrm{~m} / \mathrm{s}$. After reaching this maximum speed, his acceleration becomes zero and then he runs at constant speed. Assume that his acceleration is constant during the first $2.0 \mathrm{~s}$ of the race, that he starts from rest, and that he runs in a straight line. (a) How far has the sprinter run when he reaches his maximum speed? (b) What is the magnitude of his average velocity for a race of the following lengths: (i) $50.0 \mathrm{~m}$ (ii) $100.0 \mathrm{~m},$ (iii) $200.0 \mathrm{~m} ?$

Zachary Warner
Zachary Warner
Numerade Educator
07:16

Problem 56

On a 20 -mile bike ride, you ride the first 10 miles at an average speed of 8 $\mathrm{mi} / \mathrm{h} .$ What must your average speed over the next 10 miles be to have your average speed for the total 20 miles be (a) 4 $\mathrm{mi} / \mathrm{h} ?$ (b) 12 $\mathrm{mi} / \mathrm{h} ?$ (c) Given this average speed for the first 10 miles, can you possibly attain an average speed of 16 $\mathrm{mi} / \mathrm{h}$ for the total 20 -mile ride? Explain.

Zachary Warner
Zachary Warner
Numerade Educator
15:00

Problem 57

The position of a particle between $t=0$ and $t=2.00$ s is given by $x(t)=\left(3.00 \mathrm{m} / \mathrm{s}^{3}\right) t^{3}-\left(10.0 \mathrm{m} / \mathrm{s}^{2}\right) t^{2}+$ $(9.00 \mathrm{m} / \mathrm{s}) t$ . (a) Draw the $x-t, v_{x}-t,$ and $a_{x}-t$ graphs of this particle. (b) At what time(s) between $t=0$ and $t=2.00$ s is the particle instantaneously at rest? Does your numerical result agree with the $v_{x^{-}}$ graph in part (a)? (c) At each time calculated in part (b), is the acceleration of the particle positive or negative? Show that in each case the same answer is deduced from $a_{x}(t)$ and from the $v_{x}-t$ graph. (d) At what time(s) between $t=0$ and $t=2.00$ s is the velocity of the particle instantaneously not changing? Locate this point on the $v_{x}-t$ and $a_{x}-t$ graphs of part (a). (e) What is the particle's greatest distance from the origin $(x=0)$ between $t=0$ and $t=2.00 \mathrm{s} ?(\mathrm{f}) \mathrm{At}$ what time(s) between $t=0$ and $t=2.00$ s is the particle speeding $u p$ at the greatest rate? At what time(s) between $t=0$ and $t=2.00$ s is the particle slowing down at the greatest rate? Locate these points on the $v_{x}-t$ and $a_{x}-t$ graphs of part (a).

Zachary Warner
Zachary Warner
Numerade Educator
03:47

Problem 58

A lunar lander is descending toward the moon's surface. Until the lander reaches the surface, its height above the surface of the moon is given by $y(t)=b-c t+d t^{2},$ where $b=800 \mathrm{m}$ is the initial height of the lander above the surface, $c=60.0 \mathrm{m} / \mathrm{s},$ and $d=1.05 \mathrm{m} / \mathrm{s}^{2} .$ (a) What is the initial velocity of the lander, at $t=0 ?$ (b) What is the velocity of the lander just before it reaches the lunar surface?

Zachary Warner
Zachary Warner
Numerade Educator
01:45

Problem 59

Earthquake Analysis. Earthquakes produce several types of shock waves. The most well known are the P-waves (P for primary or pressure) and the S-waves (S for secondary or shear). In the earth's crust, the P-waves travel at around $6.5 \mathrm{km} / \mathrm{s},$ while the S-waves move at about 3.5 $\mathrm{km} / \mathrm{s} .$ The actual speeds vary depending on the type of material they are going through. The time delay between the arrival of these two waves at a seismic recording station tells gologists how far away the earthquake occurred. If the time delay is 33 s, how far from the seismic station did the earthquake occur?

Zachary Warner
Zachary Warner
Numerade Educator
02:54

Problem 60

In a relay race, each contestant runs 25.0 $\mathrm{m}$ while carrying an egg balanced on a spoon, turns around, and comes back to the starting point. Edith runs the first 25.0 m in 20.0 s. On the return trip she is more confident and takes only 15.0 s. What is the magnitude of her average velocity for (a) the first 25.0 $\mathrm{m} ?$ (b) The return trip? (c) What is her average velocity for the entire round trip? (d) What is her average speed for the round trip?

Zachary Warner
Zachary Warner
Numerade Educator
03:07

Problem 61

A rocket carrying a satellite is accelerating straight up from the earth's surface. At 1.15 s after liftoff, the rocket clears the top of its launch platform, 63 $\mathrm{m}$ above the ground. After an additional $4.75 \mathrm{s},$ it is 1.00 $\mathrm{km}$ above the ground. Calculate the magnitude of the average velocity of the rocket for (a) the 4.75 -s part of its flight and (b) the first 5.90 s of its flight.

Krystal K
Krystal K
Numerade Educator
07:07

Problem 62

The graph in Fig. P2.62 describes the acceleration as a function of time for a stone rolling down a hill starting from rest.
(a) Find the stone's velocity at $t=2.5 \mathrm{s}$ and at $t=7.5 \mathrm{s}$.
(b) Sketch a graph of the stone's velocity as a function of time.

Zachary Warner
Zachary Warner
Numerade Educator
03:07

Problem 63

Dan gets on Interstate Highway $\mathrm{I}-80$ at Seward, Nebraska, and drives due west in a straight line and at an average velocity of magnitude 88 $\mathrm{km} / \mathrm{h}$ . After traveling 76 $\mathrm{km}$ , he reaches the Aurora exit (Fig. $\mathrm{P} 2.63 ) .$ Realizing he has gone too far, he turns around and drives due east 34 $\mathrm{km}$ back to the York exit at an average velocity of magnitude 72 $\mathrm{km} / \mathrm{h}$ . For his whole trip from Seward to the York exit, what are (a) his average speed and (b) the magnitude of his average velocity?

Zachary Warner
Zachary Warner
Numerade Educator
04:43

Problem 64

A subway train starts from rest at a station and accelerates at a rate of 1.60 $\mathrm{m} / \mathrm{s}^{2}$ for 14.0 $\mathrm{s}$ . It runs at constant speed for 70.0 $\mathrm{s}$ and slows down at a rate of 3.50 $\mathrm{m} / \mathrm{s}^{2}$ until it stops at the next station. Find the total distance covered.

Zachary Warner
Zachary Warner
Numerade Educator
09:27

Problem 65

A world-class sprinter accelerates to his maximum speed in 4.0 s. He then maintains this speed for the remainder of a $100-\mathrm{m}$ race, finishing with a total time of 9.1 $\mathrm{s}$ . (a) What is the runner's average acceleration during the first 4.0 $\mathrm{s} ?$ (b) What is his average acceleration during the last 5.1 $\mathrm{s} ?$ (c) What is his average acceleration for the entire race? (d) Explain why your answer to part (c) is not the average of the answers to parts (a) and (b).

Zachary Warner
Zachary Warner
Numerade Educator
06:14

Problem 66

A sled starts from rest at the top of a hill and slides down with a constant acceleration. At some later time sled is 14.4 $\mathrm{m}$ from the top, 2.00 s after that it is 25.6 $\mathrm{m}$ from the top, 2.00 s later 40.0 $\mathrm{m}$ from the top, and 2.00 s later it is 57.6 $\mathrm{m}$ from the top. (a) What is the magnitude of the average velocity of the sled during each of the 2.00 -s intervals after passing the $14.4-\mathrm{m}$ point? (b) What is the acceleration of the sled? (c) What is the speed of the sled when it passes the $14.4-\mathrm{m}$ point? (d) How much time did it take to go from the top to the $14.4-\mathrm{m}$ point? (e) How far did the sled go during the first second after passing the $14.4-\mathrm{m}$ point?

Zachary Warner
Zachary Warner
Numerade Educator
05:47

Problem 67

A gazelle is running in a straight line (the $x$ -axis). The graph in Fig. $\mathrm{P} 2.67$ shows this animal's velocity as a function of time. During the first 12.0 $\mathrm{s}$ find (a) the total distance moved and (b) the displacement of the gazelle. (c) Sketch an $a_{x}-t$ graph showing this gazelle's acceleration as a function of time for the first 12.0 $\mathrm{s} .$

Zachary Warner
Zachary Warner
Numerade Educator
05:41

Problem 68

A rigid ball traveling in a straight line (the $x$ -axis) hits a solid wall and suddenly rebounds during a brief instant. The $v_{x}-t$ graph in Fig. $\mathrm{P} 2.68$ shows this ball's velocity as a function of time. During the first 20.0 s of its motion, find (a) the total distance the ball moves and (b) its displacement. (c) Sketch a graph of $a_{x}-t$ for this ball's motion. (d) Is the graph shown really vertical at 5.00 s? Explain.

Keshav Singh
Keshav Singh
Numerade Educator
03:35

Problem 69

A ball starts from rest and rolls down a hill with uniform acceleration, traveling 150 $\mathrm{m}$ during the second 5.0 $\mathrm{s}$ of its motion. How far did it roll during the first 5.0 s of motion?

Zachary Warner
Zachary Warner
Numerade Educator
07:37

Problem 70

Collision. The engineer of a passenger train traveling at 25.0 $\mathrm{m} / \mathrm{s}$ sights a freight train whose caboose is 200 $\mathrm{m}$ ahead onthe same track (Fig. P2.70). The freight train is traveling at 15.0 $\mathrm{m} / \mathrm{s}$ in the same direction as the passenger train. The engineer of the passenger train immediately applies the brakes, causing a constant acceleration of 0.100 $\mathrm{m} / \mathrm{s}^{2}$ in a direction opposite to the train's velocity, while the freight train continues with constant speed. Take $x=0$ at the location of the front of the passenger train when the engineer applies the brakes. (a) Will the cows nearby witness a collision? (b) If so, where will it take place? (c) On a single graph, sketch the positions of the front of the passenger train and the back of the freight train.

Zachary Warner
Zachary Warner
Numerade Educator
02:49

Problem 71

Large cockroaches can run as fast as 1.50 $\mathrm{m} / \mathrm{s}$ in short bursts. Suppose you turn on the light in a cheap motel and see one scurrying directly away from you at a constant 1.50 $\mathrm{m} / \mathrm{s} .$ If you start 0.90 m behind the cockroach with an initial speed of 0.80 $\mathrm{m} / \mathrm{s}$ toward it, what minimum constant acceleration would you need to catch up with it when it has traveled $1.20 \mathrm{m},$ just short of safety under a counter?

Zachary Warner
Zachary Warner
Numerade Educator
01:43

Problem 72

Two cars start 200 $\mathrm{m}$ apart and drive toward each other at a steady 10 $\mathrm{m} / \mathrm{s} .$ On the front of one of them, an energetic grasshopper jumps back and forth between the cars (he has strong legs!) with a constant horizontal velocity of 15 $\mathrm{m} / \mathrm{s}$ relative to the ground. The insect jumps the instant he lands, so he spends no time resting on either car. What total distance does the grasshopper travel before the cars hit?

Zachary Warner
Zachary Warner
Numerade Educator
05:27

Problem 73

An automobile and a truck start from rest at the same instant, with the automobile initially at some distance behind the truck. The truck has a constant acceleration of $2.10 \mathrm{m} / \mathrm{s}^{2},$ and the automobile an acceleration of 3.40 $\mathrm{m} / \mathrm{s}^{2} .$ The automobile overtakes the truck after the truck has moved 40.0 $\mathrm{m} .$ (a) How much time does it take the automobile to overtake the truck? (b) How far was the automobile behind the truck initially? (c) What is the speed of each when they are abreast? (d) On a single graph, sketch the position of each vehicle as a function of time. Take $x=0$ at the initial location of the truck.

Zachary Warner
Zachary Warner
Numerade Educator
06:40

Problem 74

Two stunt drivers drive directly toward each other. time $t=0$ the two cars are a distance $D$ apart, car 1 is at rest, and $\operatorname{car} 2$ is moving to the left with speed $v_{0} .$ Car 1 begins to move at $t=0,$ speeding up with a constant acceleration $a_{x}$ . Car 2 continues to move with a constant velocity. (a) At what time do the two cars collide? (b) Find the speed of car 1 just before it collides with car $2 .$ (c) Sketch $x-t$ and $v_{x}-t$ graphs for car 1 and car $2 .$ For each of the two graphs, draw the curves for both cars on the same set of axes.

Zachary Warner
Zachary Warner
Numerade Educator
02:13

Problem 75

A marble is released from one rim of a hemispherical bowl of diameter 50.0 $\mathrm{cm}$ and rolls down and up to the opposite rim in 10.0 s. Find (a) the average speed and (b) the average velocity of the marble.

Zachary Warner
Zachary Warner
Numerade Educator
04:20

Problem 76

An object's velocity is measured to be $v_{x}(t)=$ $\alpha-\beta t^{2},$ where $\alpha=4.00 \mathrm{m} / \mathrm{s}$ and $\beta=2.00 \mathrm{m} / \mathrm{s}^{3} .$ At $t=0$ the object is at $x=0 .$ (a) Calculate the object's position and acceleration as functions of time. (b) What is the object's maximum positive displacement from the origin?

Zachary Warner
Zachary Warner
Numerade Educator
05:07

Problem 77

Passing. The driver of a car wishes to pass a truck that is traveling at a constant speed of 20.0 $\mathrm{m} / \mathrm{s}$ (about 45 $\mathrm{mi} / \mathrm{h} )$ . Initially, the car is also traveling at 20.0 $\mathrm{m} / \mathrm{s}$ and its front bumper is 24.0 $\mathrm{m}$ behind the truck's rear bumper. The car accelerates at a constant 0.600 $\mathrm{m} / \mathrm{s}^{2}$ , then pulls back into the truck's lane when the rear of the car is 26.0 $\mathrm{m}$ ahead of the front of the truck. The car is 4.5 $\mathrm{m}$ long and the truck is 21.0 $\mathrm{m}$ long. (a) How much time is required for the car to pass the truck? (b) What distance does the car travel during this time? (c) What is the final speed of the car?

Supratim Pal
Supratim Pal
Numerade Educator
06:20

Problem 78

On Planet $\mathrm{X},$ you drop a 25 -kg stone from rest and measure its speed at various times. Then you use the data you obtained to construct a graph of its speed $v$ as a function of time $t$ (Fig. P2.78) From the information in the graph, answer the following questions: (a) What is $g$ on Planet $\mathrm{X} ?$ (b) An astronaut drops a piece of equipment from rest out of the landing module, 3.5 $\mathrm{m}$ above the surface of Planet $\mathrm{X}$ . How long will it take this equipment to reach the ground, and how fast will it be moving when it gets there? (c) How fast would an astronaut have to project an object straight upward to reach a height of 18.0 $\mathrm{m}$ above the release point, and how long would it take to reach that height?

Zachary Warner
Zachary Warner
Numerade Educator
05:46

Problem 79

The acceleration of a particle is given by $a_{x}(t)=$ $-2.00 \mathrm{m} / \mathrm{s}^{2}+\left(3.00 \mathrm{m} / \mathrm{s}^{3}\right) t .$ (a) Find the initial velocity $v_{0 x}$ such that the particle will have the same $x$ -coordinate at $t=4.00 \mathrm{s}$ as it had at $t=0 .$ (b) What will be the velocity at $t=4.00$ s?

Zachary Warner
Zachary Warner
Numerade Educator
02:41

Problem 80

Egg Drop. You are on the roof of the physics building, 46.0 $\mathrm{m}$ above the ground (Fig. $\mathrm{P} 2.80 \mathrm{m} .$ Your physics professor, who is 1.80 $\mathrm{m}$ tall, is walking alongside the building at a constant speed of 1.20 $\mathrm{m} / \mathrm{s} .$ If you wish to drop an egg on your professor's head, where should the the egg? Assume that the egg is
in free fall.

Zachary Warner
Zachary Warner
Numerade Educator
03:20

Problem 81

$\mathrm{A}$ certain volcano on earth can eject rocks vertically to a maximum height $H .$ (a) How high (in terms of $H$ ) would these rocks go if a volcano on Mars ejected them with the same initial velocity? The acceleration due to gravity on Mars is $3.71 \mathrm{m} / \mathrm{s}^{2},$ and you can neglect air resistance on both planets. (b) If the rocks are in the air for a time $T$ on earth, for how long (in terms of $T$) will they be in the air on Mars?

Zachary Warner
Zachary Warner
Numerade Educator
03:36

Problem 82

An entertainer juggles balls while doing other activities. In one act, she throws a ball vertically upward, and while it is in the air, she runs to and from a table 5.50 $\mathrm{m}$ away at a constant speed of 2.50 $\mathrm{m} / \mathrm{s}$ , returning just in time to catch the falling ball. (a) With what minimum initial speed must she throw the ball upward to accomplish this feat? (b) How high above its initial position is the ball just as she reaches the table?

Zachary Warner
Zachary Warner
Numerade Educator
03:05

Problem 83

Visitors at an amusement park watch divers step off a platform 21.3 $\mathrm{m}(70 \mathrm{ft})$ above a pool of water. According to the announcer, the divers enter the water at a speed of 56 $\mathrm{mi} / \mathrm{h}$ $(25 \mathrm{m} / \mathrm{s}) \cdot$ Air resistance may be ignored. (a) Is the announcer correct in this claim? (b) Is it possible for a diver to leap directly upward off the board so that, missing the board on the way down, she enters the water at 25.0 $\mathrm{m} / \mathrm{s} ?$ If so, what initial upward speed is required? Is the required initial speed physically attainable?

Zachary Warner
Zachary Warner
Numerade Educator
03:24

Problem 84

A flowerpot falls off a windowsill and falls past the window below. You may ignore air resistance. It takes the pot 0.420 s to pass from the top to the bottom of this window, which is 1.90 $\mathrm{m}$ high. How far is the top of the window below the windowsill from which the flowerpot fell?

Zachary Warner
Zachary Warner
Numerade Educator
05:58

Problem 85

Look Out Below. Sam heaves a 16-lb shot straight upward, giving it a constant upward acceleration from rest of 35.0 $\mathrm{m} / \mathrm{s}^{2}$ for 64.0 $\mathrm{cm} .$ He releases it 2.20 $\mathrm{m}$ above the ground. You may ignore air resistance. (a) What is the speed of the shot when Sam releases it? (b) How high above the ground does it go? (c) How much time does he have to get out of its way before it returns to the height of the top of his head, 1.83 m above the ground?

Zachary Warner
Zachary Warner
Numerade Educator
07:43

Problem 86

A Multistage Rocket. In the first stage of a two-stage rocket, the rocket is fired from the launch pad starting from rest but with a constant acceleration of 3.50 $\mathrm{m} / \mathrm{s}^{2}$ upward. At 25.0 s after launch, the second stage fires for $10.0 \mathrm{s},$ which boosts the rocket's velocity to 132.5 $\mathrm{m} / \mathrm{s}$ upward at 35.0 s after launch. This firing uses up all the fuel, however, so after the second stage has finished firing, the only force acting on the rocket is gravity. Air resistance can be neglected. (a) Find the maximum height that the stage-two rocket reaches above the launch pad. (b) How much time after the end of the stage-two firing will it take for the rocket to fall back to the launch pad? (c) How fast will the stage-two rocket be moving just as it reaches the launch pad?

Zachary Warner
Zachary Warner
Numerade Educator
06:30

Problem 87

Juggling Act. A juggler performs in a room whose ceiling is 3.0 $\mathrm{m}$ above the level of his hands. He throws a ball upward so that it just reaches the ceiling. (a) What is the initial velocity of the ball? (b) What is the time required for the ball to reach the ceiling? At the instant when the first ball is at the ceiling, the juggler throws a second ball upward with two-thirds the initial velocity of the first. (c) How long after the second ball is thrown do the two balls pass each other? (d) At what distance above the juggler's hand do they pass each other?

Zachary Warner
Zachary Warner
Numerade Educator
03:32

Problem 88

A physics teacher performing an outdoor demonstration suddenly falls from rest off a high cliff and simultaneously shouts "Help." When she has fallen for 3.0 s, she hears the echo of her shout from the valley floor below. The speed of sound is 340 $\mathrm{m} / \mathrm{s}$. (a) How tall is the cliff? (b) If air resistance is neglected, how fast will she be moving just before she hits the ground? (Her actual speed will be less than this, due to air resistance.)

Zachary Warner
Zachary Warner
Numerade Educator
08:20

Problem 89

A helicopter carrying Dr. Evil takes off with a constant upward acceleration of 5.0 $\mathrm{m} / \mathrm{s}^{2}$ . Secret agent Austin Powers jumps on just as the helicopter lifts off the ground. After the two men struggle for 10.0 s, Powers shuts off the engine and steps out of the helicopter. Assume that the helicopter is in free fall after its engine is shut off, and ignore the effects of air resistance. (a) What is the maximum height above ground reached by the helicopter? (b) Powers deploys a jet pack strapped on his back 7.0 s after leaving the helicopter, and then he has a constant downward acceleration with magnitude 2.0 $\mathrm{m} / \mathrm{s}^{2} .$ How far is Powers above the ground when the helicopter crashes into the ground?

Zachary Warner
Zachary Warner
Numerade Educator
07:29

Problem 90

Cliff Height. You are climbing in the High Sierra where you suddenly find yourself at the edge of a fog-shrouded cliff. To find the height of this cliff, you drop a rock from the top and 10.0 $\mathrm{s}$ later hear the sound of it hitting the ground at the foot of the cliff. (a) Ignoring air resistance, how high is the cliff if the speed of sound is 330 $\mathrm{m} / \mathrm{s} ?$ (b) Suppose you had ignored the time it takes the sound to reach you. In that case, would you have overestimated or underestimated the height of the cliff? Explain your reasoning.

Zachary Warner
Zachary Warner
Numerade Educator
04:25

Problem 91

Falling Can. A painter is standing on scaffolding that is raised at constant speed. As he travels upward, he accidentally nudges a paint can off the scaffolding and it falls 15.0 $\mathrm{m}$ to the ground. You are watching, and measure with your stopwatch that it takes 3.25 s for the can to reach the ground. Ignore air resistance. (a) What is the speed of the can just just before it hits the ground? (b) Another painter is standing on a ledge, with his hands 4.00 $\mathrm{m}$ above the can when it falls off. He has lightning-fast reflexes and if the can passes in front of him, he can catch it. Does he get the chance?

Zachary Warner
Zachary Warner
Numerade Educator
05:10

Problem 92

Determined to test the law of gravity for himself, a student walks off a skyscraper 180 $\mathrm{m}$ high, stopwatch in hand, and starts his free fall (zero initial velocity). Five seconds later, Superman arrives at the scene and dives off the roof to save the student. Superman leaves the roof with an initial speed $v_{0}$ that he produces by pushing himself downward from the edge of the roof with his legs of steel. He then falls with the same acceleration as any freely falling body. (a) What must the value of $v_{0}$ be so that Superman catches the student just before they reach the ground? (b) On the same graph, sketch the positions of the student and of Superman as functions of time. Take Superman's initial speed to have the value calculated in part (a). (c) If the height of the skyscraper is less than some minimum value, even Superman can't reach the student before he hits the ground. What is this minimum height?

Zachary Warner
Zachary Warner
Numerade Educator
07:32

Problem 93

During launches, rockets often discard unneeded parts. A certain rocket starts from rest on the launch pad and accelerates upward at a steady 3.30 $\mathrm{m} / \mathrm{s}^{2}$ . When it is 235 $\mathrm{m}$ above the launch pad, it discards a used fuel canister by simply disconnecting it. Once it is disconnected, the only force acting on the canister is gravity (air resistance can be ignored). (a) How high is the rocket when the canister hits the launch pad, assuming that the rocket does not change its acceleration? (b) What total distance did the canister travel between its release and its crash onto the launch pad?

Zachary Warner
Zachary Warner
Numerade Educator
02:41

Problem 94

A ball is thrown straight up from the ground with speed $v_{0} .$ At the same instant, a second ball is dropped from rest from a height $H,$ directly above the point where the first ball was thrown upward. There is no air resistance. (a) Find the time at which the two balls collide. (b) Find the value of $H$ in terms of $v_{0}$ and $g$ so that at the instant when the balls collide, the first ball is at the highest point of its motion.

Zachary Warner
Zachary Warner
Numerade Educator
07:33

Problem 95

Two cars, $A$ and $B,$ travel in a straight line. The dis tance of $A$ from the starting point is given as a function of time by $x_{A}(t)=\alpha t+\beta t^{2},$ with $\alpha=2.60 \mathrm{m} / \mathrm{s}$ and $\beta=1.20 \mathrm{m} / \mathrm{s}^{2} .$ The distance of $B$ from the starting point is $x_{B}(t)=\gamma t^{2}-\delta t^{3},$ with $\gamma=2.80 \mathrm{m} / \mathrm{s}^{2}$ and $\delta=0.20 \mathrm{m} / \mathrm{s}^{3} .$ (a) Which car is ahead just after they leave the starting point? (b) At what time(s) are the cars at the same point? (c) At what time(s) is the distance from $A$ to $B$ neither increasing nor decreasing? (d) At what time(s) do $A$ and $B$ have the same acceleration?

Zachary Warner
Zachary Warner
Numerade Educator
05:05

Problem 96

In the vertical jump, an athlete starts from a crouch and jumps upward to reach as high as possible. Even the best athletes spend little more than 1.00 s in the air (their "hang time"). Treat the athlete as a particle and let $y_{\text { max }}$ be his maximum height above the floor. To explain why he seems to hang in the air, calculate the ratio of the time he is above $y_{\max } / 2$ to the time it takes him to go from the floor to that height. You may ignore air resistance.

Zachary Warner
Zachary Warner
Numerade Educator
08:22

Problem 97

Catching the Bus. A student is running at her top speed of 5.0 $\mathrm{m} / \mathrm{s}$ to catch a bus, which is stopped at the bus stop. When the student is still 40.0 $\mathrm{m}$ from the bus, it starts to pull away, moving with a constant acceleration of 0.170 $\mathrm{m} / \mathrm{s}^{2}$ . (a) For how much time and what distance does the student have to run at 5.0 $\mathrm{m} / \mathrm{s}$ before she overtakes the bus? (b) When she reaches the bus, how fast is the bus traveling? (c) Sketch an $x$ -t graph for both the student and the bus. Take $x=0$ at the initial position of the student. (d) The equations you used in part (a) to find the time have a second solution, corresponding to a later time for which the student and bus are again at the same place if they continue their specified motions. Explain the significance of this second solution. How fast is the bus traveling at this point? (e) If the student's top speed is 3.5 $\mathrm{m} / \mathrm{s},$ will she catch the bus? (f) What is the minimum speed the student must have to just catch up with the bus? For what time and what distance does she have to run in that case?

Zachary Warner
Zachary Warner
Numerade Educator
06:01

Problem 98

An alert hiker sees a boulder fall from the top of a distant cliff and notes that it takes 1.30 s for the boulder to fall the last third of the way to the ground. You may ignore air resistance. (a) What is the height of the cliff in meters? (b) If in part (a) you get two solutions of a quadratic equation and you use one for your answer, what does the other solution represent?

Zachary Warner
Zachary Warner
Numerade Educator
12:12

Problem 99

A ball is thrown straight up from the edge of the roof of a building. A second ball is dropped from the roof 1.00 s later. You may ignore air resistance. (a) If the height of the building is $20.0 \mathrm{m},$ what must the initial speed of the first ball be if both are to hit the ground at the same time? On the same graph, sketch the position of each ball as a function of time, measured from when the first ball is thrown. Consider the same situation, but now let the initial speed $v_{0}$ of the first ball be given and treat the height $h$ of the building as an unknown. (b) What must the height of the building be for both balls to reach the ground at the same time (i) if $v_{0}$ is 6.0 $\mathrm{m} / \mathrm{s}$ and (ii) if $v_{0}$ is 9.5 $\mathrm{m} / \mathrm{s} ?$ (c) If $v_{0}$ is greater than some value $v_{\text { max }},$ a value of $h$ does not exist that allows both balls to hit the ground at the same time. Solve for $v_{\text { max. }}$ . The value $v_{\text { max }}$ has a simple physical interpretation. What is it? (d) If $v_{0}$ is less than some value $v_{\text { min }}$ , a value of $h$ does not exist that allows both balls to hit the ground at the same time. Solve for $v_{\text { min. }}$ The value $v_{\min }$ also has a simple physical interpretation. What is it?

Zachary Warner
Zachary Warner
Numerade Educator