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College Physics 2017

Raymond A. Serway, Chris Vuille, John hughes

Chapter 3

Motion in two Dimensions

Educators


Problem 1

An airplane in a holding pattern flies at constant altitude along a circular path of radius 3.50 $\mathrm{km}$ . If the airplane rounds half the circle in $1.50 \times 10^{2} \mathrm{s},$ determine the magnitude of its (a) displacement and (b) average velocity during that time. (c) What is the airplane's average speed during the same time interval?

Averell H.
Carnegie Mellon University

Problem 2

A hiker walks 2.00 $\mathrm{km}$ north and then 3.00 $\mathrm{km}$ east, all in 2.50 hours. Calculate the magnitude and direction of the hiker's (a) displacement (in $\mathrm{km} )$ and (b) average velocity (in $\mathrm{km} / \mathrm{h}$ ) during those 2.50 hours. (c) What was her average speed during the same time interval?

Averell H.
Carnegie Mellon University

Problem 3

A miniature quadcopter is located at $x_{i}=2.00 \mathrm{m}$ and $y_{i}=$ 4.50 $\mathrm{m}$ at $t=0$ and moves with an average velocity having components $v_{\mathrm{av}, x}=1.50 \mathrm{m} / \mathrm{s}$ and $v_{\mathrm{ax}, y}=-1.00 \mathrm{m} / \mathrm{s}$ . What are the (a) $x$ -coordinate and (b) y-coordinate of the quadcopter's position at $t=2.00 \mathrm{s}$ ?

Averell H.
Carnegie Mellon University

Problem 4

An ant crawls on the floor along the curved path shown in Figure $P 3.4$ . The ant's positions and velocities are indicated for times $t_{i}=0$ and $t_{f}=$ 5.00 s. Determine the $x-$ and $y$ -components of the ant's (a) displacement, (b) average velocity, and (c) average acceleration between the two times.

Averell H.
Carnegie Mellon University

Problem 5

A car is traveling east at 25.0 $\mathrm{m} / \mathrm{s}$ when it turns due north and accelerates to $35.0 \mathrm{m} / \mathrm{s},$ all during a time of 6.00 s. Calculate the magnitude of the car's average acceleration.

Averell H.
Carnegie Mellon University

Problem 6

A rabbit is moving in the positive $x$ -direction at 2.00 $\mathrm{m} / \mathrm{s}$ when it spots a predator and accelerates to a velocity of 12.0 $\mathrm{m} / \mathrm{s}$ along the negative $y$ -axis, all in 1.50 $\mathrm{s}$ . Determine (a) the $x$ -component and (b) the $y$ -component of the rabbit's acceleration.

Averell H.
Carnegie Mellon University

Problem 7

A student stands at the edge of a cliff and throws a stone horizontally over the edge with a speed of 18.0 $\mathrm{m} / \mathrm{s}$ The cliff is 50.0 $\mathrm{m}$ above a flat, horizontal beach as shown in Figure $\mathrm{P} 3.7$ . (a) What are the coordinates of the initial position of the stone? (b) What are
the components of the initial velocity? (c) Write the equations for the $x$ - and $y$ -components of the velocity of the stone with time. (d) Write the equations for the position of the stone with time, using the coordinates in Figure P3.7. (e) How long after being released does the stone strike the beach below the cliff? (f) With what speed and angle of impact does the stone land?

Averell H.
Carnegie Mellon University

Problem 8

One of the fastest recorded pitches in major league bascball, thrown by Tim Lincecum in 2009 , was clocked at 101.0 $\mathrm{mi} / \mathrm{h}$ (Fig. P3.8). If a pitch were thrown horizontally with this velocity, how far would the ball fall vertically by the time it reached home plate, 60.5 $\mathrm{ft}$ away?

Averell H.
Carnegie Mellon University

Problem 9

The best leaper in the animal kingdom is the puma, which can jump to a height of 3.7 $\mathrm{m}$ when leaving the ground at an angle of $45^{\circ} .$ With what speed must the animal leave the ground to reach that height?

Averell H.
Carnegie Mellon University

Problem 10

A rock is thrown upward from the level ground in such a way that the maximum height of its flight is equal to its horizontal range $R$ . (a) At what angle $\theta$ is the rock thrown? (b) In terms of the original range $R$ , what is the range $R_{\max }$ the rock can attain if it is launched at the same speed but at the optimal angle for maximum range? (c) Would your answer to part (a) be different if the rock is thrown with the same speed on a different planet? Explain.

Averell H.
Carnegie Mellon University

Problem 11

A placekicker must kick a football from a point 36.0 $\mathrm{m}$ (about 40 yards) from the goal. Half the crowd hopes the ball will clear the crossbar, which is 3.05 $\mathrm{m}$ high. When kicked, the ball leaves the ground with a speed of 20.0 $\mathrm{m} / \mathrm{s}$ at an angle of $53.0^{\circ}$ to the horizontal. (a) By how much does the ball clear or fall short of clearing the crossbar? (b) Does the ball approach the crossbar while still rising or while falling?

Averell H.
Carnegie Mellon University

Problem 12

The record distance in the sport of throwing cowpats is 81.1 $\mathrm{m} .$ This record toss was set by Steve Urner of the United States in 1981 . Assuming the initial launch angle was $45^{\circ}$ and neglecting air resistance, determine (a) the initial speed of the projectile and (b) the total time the projectile was in flight. (c) Qualitatively, how would the answers change if the launch angle were greater than $45^{\circ}$ ? Explain.

Averell H.
Carnegie Mellon University

Problem 13

A brick is thrown upward from the top of a building at an angle of $25^{\circ}$ to the horizontal and with an initial speed of 15 $\mathrm{m} / \mathrm{s}$ . If the brick is in flight for $3.0 \mathrm{s},$ how tall is the building?

Averell H.
Carnegie Mellon University

Problem 14

From the window of a building, a ball is tossed from a height $y_{0}$ above the ground with an initial velocity of 8.00 $\mathrm{m} / \mathrm{s}$ and angle of $20.0^{\circ}$ below the horizontal. It strikes the ground 3.00 s later. (a) If the base of the building is taken to be the origin of the coordinates, with upward the positive $y$ -direction, what are the initial coordinates of the ball? (b) With the positive $x$ -direction chosen to be out the window, find the $x$ - and $y$ -components of the initial velocity. (c) Find the equations for the $x$ - and $y$ -components of the position as functions of time. (d) How far horizontally from the base of the building does the ball strike the ground? (e) Find the height from which the ball was thrown. (f) How long does it take the ball to reach a point 10.0 $\mathrm{m}$ below the level of launching?

Averell H.
Carnegie Mellon University

Problem 15

A car is parked on a cliff overlooking the ocean on an incline that makes an angle of $24.0^{\circ}$ below the horizontal. The negligent driver leaves the car in neutral, and the emergency brakes are defective. The car rolls from rest down the incline with a constant acceleration of 4.00 $\mathrm{m} / \mathrm{s}^{2}$ for a distance of 50.0 $\mathrm{m}$ to the edge of the cliff, which is 30.0 $\mathrm{m}$ above the ocean. Find (a) the car's position relative to the base of the cliff when the car lands in the ocean and (b) the length of time the car is in the air.

Averell H.
Carnegie Mellon University

Problem 16

An artillery shell is fired with an initial velocity of 300 $\mathrm{m} / \mathrm{s}$ at $55.0^{\circ}$ above the horizontal. To clear an avalanche, it explodes on a mountainside 42.0 $\mathrm{s}$ after firing. What are the $x-$ and $y$ -coordinates of the shell where it explodes, relative to its firing point?

Averell H.
Carnegie Mellon University

Problem 17

A projectile is launched with an initial speed of 60.0 $\mathrm{m} / \mathrm{s}$ at an angle of $30.0^{\circ}$ above the horizontal. The projectile lands on a hillside 4.00 s later. Neglect air friction. (a) What is the projectile's velocity at the highest point of its trajectory? (b) What is the straight-line distance from where the projectile was launched to where it hits its target?

Averell H.
Carnegie Mellon University

Problem 18

A fireman $d=50.0 \mathrm{m}$ away from a burning building directs a stream of water from a ground-level fire hose at an angle of $\theta_{i}=30.0^{\circ}$ above the horizontal as shown in Figure $P 3.18$ . If the speed of the stream as it leaves the hose is $v_{i}=40.0 \mathrm{m} / \mathrm{s},$ at what height will the stream of water strike the building?

Averell H.
Carnegie Mellon University

Problem 19

A playground is on the flat roof of a city school, 6.00 $\mathrm{m}$ above the street below (Fig. $\mathrm{P} 3.19 ) .$ The vertical wall of the building is $h=7.00 \mathrm{m}$ high, to form a 1 -m-high railing around the playground. A ball has fallen to the street below, and a passerby returns it by launching it at an angle of $\theta=53.0^{\circ}$ above the horizontal at a point $d=24.0 \mathrm{m}$ from the base of the building wall. The ball takes 2.20 $\mathrm{s}$ to reach a point vertically above the wall. (a) Find the speed at which the ball was launched. (b) Find the vertical distance by which the ball clears the wall. (c) Find the horizontal distance from the wall to the point on the roof where the ball lands.

Averell H.
Carnegie Mellon University

Problem 20

A cruise ship sails due north at 4.50 $\mathrm{m} / \mathrm{s}$ while a Coast Guard patrol boat heads $45.0^{\circ}$ north of west at 5.20 $\mathrm{m} / \mathrm{s}$ . What are (a) the $x$ -component and $(\mathrm{b})$ y-component of the velocity of the cruise ship relative to the patrol boat?

Averell H.
Carnegie Mellon University

Problem 21

Suppose a boat moves at 12.0 $\mathrm{m} / \mathrm{s}$ relative to the water. If the boat is in a river with the current directed east at $2.50 \mathrm{m} / \mathrm{s},$ what is the boat's speed relative to the ground when it is heading (a) east, with the current, and (b) west, against the current?

Averell H.
Carnegie Mellon University

Problem 22

A car travels due east with a speed of 50.0 $\mathrm{km} / \mathrm{h}$. Raindrops are falling at a constant speed vertically with respect to the Farth. The traces of the rain on the side windows of the car make an angle of $60.0^{\circ}$ with the vertical. Find the velocity of the rain with respect to (a) the car and (b) the Earth.

Averell H.
Carnegie Mellon University

Problem 23

A jet airliner moving initially at $3.00 \times 10^{2} \mathrm{mi} / \mathrm{h}$ due east enters a region where the wind is blowing $1.00 \times 10^{2} \mathrm{mi} / \mathrm{h}$ in a direction $30.0^{\circ}$ north of east. (a) Find the components of the velocity of the jet airliner relative to the air, $\overrightarrow{\mathbf{v}}_{\mathrm{jA}}$ . (b) Find the components of the velocity of the air relative to Earth, $\overrightarrow{\mathbf{v}}_{\mathrm{AF}}(\mathrm{c})$ Write an equation analogous to Equation 3.11 for the velocities $\overrightarrow{\mathbf{v}}_{\mathrm{JA}}, \overrightarrow{\mathbf{v}}_{\mathrm{AE}},$ and $\overrightarrow{\mathbf{v}}_{\mathrm{JE}}$ (d) What are the speed and direction of the aircraft relative to the ground?

Averell H.
Carnegie Mellon University

Problem 24

A Coast Guard cutter detects an unidentified ship at a distance of 20.0 $\mathrm{km}$ in the direction $15.0^{\circ}$ east of north. The ship is traveling at 26.0 $\mathrm{km} / \mathrm{h}$ on a course at $40.0^{\circ}$ east of north. The Coast Guard wishes to send a speedboat to intercept and investigate the vessel. (a) If the speedboat travels at 50.0 $\mathrm{km} / \mathrm{h},$ in what direction should it head? Express the direction $\mathrm{as}$ a compass bearing with respect to due north. (b) Find the time required for the cutter to intercept the ship.

Averell H.
Carnegie Mellon University

Problem 25

A bolt drops from the ceiling of a moving train car that is accelerating northward at a rate of 2.50 $\mathrm{m} / \mathrm{s}^{2}$ . (a) What is the acceleration of the bolt relative to the train car' (b) What is the acceleration of the bolt relative to the Earth? (c) Describe the trajectory of the bolt as seen by an observer fixed on the Earth.

Keshav S.
Numerade Educator

Problem 26

In An airplane maintains a speed of 630 $\mathrm{km} / \mathrm{h}$ relative to the air it is flying through, as it makes a trip to a city 750 $\mathrm{km}$ away to the north. (a) What time interval is required for the trip if the plane flies through a headwind blowing at 35.0 $\mathrm{km} / \mathrm{h}$ toward the south? (b) What time interval is required if there is a tailwind with the same speed? (c) What time interval is required if there is a crosswind blowing at 35.0 $\mathrm{km} / \mathrm{h}$ the east relative to the ground?

Averell H.
Carnegie Mellon University

Problem 27

Suppose a chinook salmon needs to jump a waterfall that is 1.50 $\mathrm{m}$ high. If the fish starts from a distance 1.00 $\mathrm{m}$ from the base of the ledge over which the waterfall flows, (a) find the $x$ - and $y$ -components of the initial velocity the salmon would need to just reach the ledge at the top of its trajectory. (b) Can the fish make this jump? (Note that a chinook salmon can jump out of the water with an initial speed of 6.26 $\mathrm{m} / \mathrm{s} .$ )

Averell H.
Carnegie Mellon University

Problem 28

A bomber is flying horizontally over level terrain at a speed of 275 $\mathrm{m} / \mathrm{s}$ relative to the ground and at an altitude of 3.00 $\mathrm{km}$ . (a) The bombardier releases one bomb. How far does the bomb travel horizontally between its release and its impact on the ground? Ignore the effects of air resistance. (b) Firing from the people on the ground suddenly incapacitates the bombardier before he can call, "Bombs away!" Consequently, the pilot maintains the plane's original course, altitude, and speed through a storm of flak. Where is the plane relative to the bomb's point of impact when the bomb hits the ground? (c) The plane has a telescopic bomb sight set so that the bomb hits the target seen in the sight at the moment of release. At what angle from the vertical was the bombsight set?

Averell H.
Carnegie Mellon University

Problem 29

A river has a steady speed of 0.500 $\mathrm{m} / \mathrm{s}$ . A student swims upstream a distance of 1.00 $\mathrm{km}$ and swims back to the starting point. (a) If the student can swim at a speed of 1.20 $\mathrm{m} / \mathrm{s}$ in still water, how long does the trip take? (b) How much time is required in still water for the same length swim? (c) Intuitively, why does the swim take longer when there is a current?

Yaqub K.
Numerade Educator

Problem 30

This is a symbolic version of Problem 29. A river has a steady speed of $v_{s}$ A student swims upstream a distance $d$ and back to the starting point. (a) If the student can swim at a speed of $v$ in still water, how much time $t_{\text { up }}$ does it take the student to swim upstream a distance $d ?$ Express the answer in terms of $d, v,$ and $v_{s .}$ (b) Using the same variables, how much time $t_{\text { down }}$ does it take to swim back downstream to the starting point? (c) Sum the answers found in parts (a) and (b) and show that the time $t_{a}$ required for the whole trip can be written as
$$t_{a}=\frac{2 d / v}{1-v_{s}^{2} / v^{2}}$$
(d) How much time $t_{b}$ does the trip take in still water?
(e) Which is larger, $t_{a}$ or $t_{b}^{2}$ Is it always larger?

Laszlo Z.
Numerade Educator

Problem 31

How long does it take an automobile traveling in the left lane of a highway at 60.0 $\mathrm{km} / \mathrm{h}$ to overtake (become even with) another car that is traveling in the right lane at 40.0 $\mathrm{km} / \mathrm{h}$ when the cars' front bumpers are initially 100 $\mathrm{m}$ apart?

Averell H.
Carnegie Mellon University

Problem 32

A moving walkway at an airport has a speed $v_{1}$ and a length $L$ . A woman stands on the walkway as it moves from one end to the other, while a man in a hurry to reach his flight walks on the walkway with a speed of $v_{2}$ relative to the moving walkway. (a) How long does it take the woman to travel the distance $I Z$ (b) How long does it take the man to travel this distance?

Averell H.
Carnegie Mellon University

Problem 33

A boy throws a baseball onto a roof and it rolls back down and off the roof with a speed of 3.75 $\mathrm{m} / \mathrm{s}$ . If the roof is pitched at $35.0^{\circ}$ below the horizon and the roof edge is 2.50 $\mathrm{m}$ above the ground, find (a) the time the baseball spends in the air, and (b) the horizontal distance from the roof edge to the point where the baseball lands on the ground.

Averell H.
Carnegie Mellon University

Problem 34

You can use any coordinate system you like to solve a projectile motion problem. To demonstrate the truth of this statement, consider a ball thrown off the top of a building with a velocity $\overrightarrow{\mathbf{v}}$ at an angle $\theta$ with respect to the horizontal. Let the building be 50.0 $\mathrm{m}$ tall, the initial horizontal velocity be 9.00 $\mathrm{m} / \mathrm{s}$ , and the initial vertical velocity be 12.0 $\mathrm{m} / \mathrm{s}$ . Choose your coordinates such that the positive $y$ -axis is upward, the $x$ -axis is to the right, and the origin is at the point where the ball is released. (a) With these choices, find the ball's maximum height above the ground and the time it takes to reach the maximum height. (b) Repeat your calculations choosing the origin at the base of the building.

Check back soon!

Problem 35

Towns $A$ and $B$ in Figure $P 3.35$ are 80.0 km apart. A couple arranges to drive from town $A$ and meet a couple driving from town $\mathrm{B}$ at the lake, $\mathrm{L}$ . The two couples leave simultaneously and drive for 2.50 $\mathrm{h}$ in the directions shown. Car 1 has a speed of 90.0 $\mathrm{km} / \mathrm{h}$ . If the cars arrive simultaneously at the lake, what is the speed of car 2 ?

Averell H.
Carnegie Mellon University

Problem 36

In is In a local diner, a customer slides an empty coffee cup down the counter for a refill. The cup slides off the counter and strikes the floor at distance $d$ from the base of the counter. If the height of the counter is $h,(\text { a })$ find an expression for the time $t$ it takes the cup to fall to the floor in terms of the variables $h$ and $g$ . (b) With what speed does the mug leave the counter? Answer in terms of the variables $d, g,$ and $h .(\mathrm{c})$ In the same terms, what is the speed of the cup immediately before it hits the floor? (d) In terms of $h$ and $d$ , what is the direction of the cup's velocity immediately before it hits the floor?

Averell H.
Carnegie Mellon University

Problem 37

A father demonstrates projectile motion to his children by placing a pea on his fork's handle and rapidly depressing the curved tines, launching the pea to heights above the dining room table. Suppose the pea is launched at 8.25 $\mathrm{m} / \mathrm{s}$ at an angle of $75.0^{\circ}$ above the table. With what speed does the pea strike the ceiling 2.00 $\mathrm{m}$ above the table?

Averell H.
Carnegie Mellon University

Problem 38

Two canoeists in identical canoes exert the same effort paddling and hence maintain the same speed relative to the water. One paddles directly upstream (and moves upstream), whereas the other paddles directly downstream. With downstream as the positive direction, an observer on shore determines the velocities of the two canoes to be $-1.2 \mathrm{m} / \mathrm{s}$ and $+2.9 \mathrm{m} / \mathrm{s}$ , respecties of the two canoes to be $-1.2 \mathrm{m} / \mathrm{s}$ and $+2.9 \mathrm{m} / \mathrm{s}$ , respectively. (a) What is the speed of the water relative to the shore? (b) What is the speed of each canoe relative to the water?

Averell H.
Carnegie Mellon University

Problem 39

A rocket is launched at an angle of $53.0^{\circ}$ above the horizontal with an initial speed of $100 . \mathrm{m} / \mathrm{s}$ . The rocket moves for 3.00 $\mathrm{s}$ along its initial line of motion with an acceleration of 30.0 $\mathrm{m} / \mathrm{s}^{2}$ . At this time, its engines fail and the rocket proceeds to move as a projectile. Find (a) the maximum altitude reached by the rocket, (b) its total time of flight, and (c) its horizontal range.

Check back soon!

Problem 40

A farm truck travels due east with a constant speed of 9.50 $\mathrm{m} / \mathrm{s}$ along a horizontal road. A boy riding in the back of the truck tosses a can of soda upward (Fig. P3.40) and catches it at the same location in the truck bed, but 16.0 $\mathrm{m}$ farther down the road. Ignore any effects of air resistance, (a) At what angle to the vertical does the boy throw the can, relative to the moving truck? (b) What is the can's initial speed relative to the truck? (c) What is the shape of the can's trajectory as seen by the boy? (d) What is the shape of the can's trajectory as seen by a stationary observer on the ground? (e) What is the initial velocity of the can, relative to the stationary observer?

Averell H.
Carnegie Mellon University

Problem 41

(a) If a person can jump a maximum horizontal distance (by using a $45^{\circ}$ projection angle $)$ of 3.0 $\mathrm{m}$ on Earth, what would be his maximum range on the Moon, where the free-fall acceleration is $g / 6$ and $g=9.80 \mathrm{m} / \mathrm{s}^{2}$ (b) Repeat for Mars, where the acceleration due to gravity is 0.38$g$ .

Averell H.
Carnegie Mellon University

Problem 42

A ball is thrown straight upward and returns to the thrower's hand after 3.00 s in the air. A second ball thrown at an angle of $30.0^{\circ}$ with the horizontal reaches the same maximum height as the first ball. (a) At what speed was the first ball thrown? (b) At what speed was the second ball thrown?

Averell H.
Carnegie Mellon University

Problem 43

A home run is hit in such a way that the baseball just clears a wall 21 $\mathrm{m}$ high, located 130 $\mathrm{m}$ from home plate. The ball is hit at an angle of $35^{\circ}$ to the horizontal, and air resistance is negligible. Find (a) the initial speed of the ball, (b) the time it takes the ball to reach the wall, and (c) the velocity components and the speed of the ball when it reaches the wall. (Assume the ball is hit at a height of 1.0 $\mathrm{m}$ above the ground.)

Averell H.
Carnegie Mellon University

Problem 44

A $2.00-\mathrm{m}$ -tall basketball player is standing on the floor 10.0 $\mathrm{m}$ from the basket, as in Figure $\mathrm{P} 3.44$ . If he shoots the ball at a $40.0^{\circ}$ angle with the horizontal, at what initial speed must he throw the basketball so that it goes through the hoop without striking the backboard? The height of the basket is 3.05 $\mathrm{m} .$

Averell H.
Carnegie Mellon University

Problem 45

A quarterback throws a football toward a receiver with an initial speed of $20 . \mathrm{m} / \mathrm{s}$ at an angle of $30 .$ above the horizontal. At that instant the receiver is $20 . \mathrm{m}$ from the quarterback. In (a) what direction and (b) with what constant speed should the receiver run in order to catch the football at the level at which it was thrown?

Averell H.
Carnegie Mellon University

Problem 46

The $x$ - and $y$ -coordinates of a projectile launched from the origin are $x=v_{0 x} t$ and $y=v_{0 y} t-\frac{1}{2} g t^{2}$ . Solve the first of these equations for time $t$ and substitute into the second to show that the path of a projectile is a parabola with the form $y=a x+b x^{2},$ where $a$ and $b$ are constants.

Averell H.
Carnegie Mellon University

Problem 47

Spittingcobras can defend themselves by squeezing muscles around their venom glands to squirt venom at an attacker. Suppose a spitting cobra rears up to a height of 0.500 $\mathrm{m}$ above the ground and launches venom at 3.50 $\mathrm{m} / \mathrm{s}$ , directed $50.0^{\circ}$ above the horizon. Neglecting air resistance, find the horizontal distance traveled by the venom before it hits the ground.

Averell H.
Carnegie Mellon University

Problem 48

When baseball outfielders throw the ball, they usually allow it to take one bounce, on the theory that the ball arrives at its target sooner that way. Suppose that, after the bounce, the ball rebounds at the same angle $\theta$ that it had when it was released (as in Fig. P9.48), but loses half its speed. (a) Assuming that the ball is always thrown with the same initial speed, at what angle $\theta$ should the ball be thrown in order to go the same distance $D$ with one bounce as a ball thrown upward at $45.0^{\circ}$ with no bounce? (b) Determine the ratio of the times for the one-bounce and no-bounce throws.

Averell H.
Carnegie Mellon University

Problem 49

A hunter wishes to cross a river that is 1.5 $\mathrm{km}$ wide and flows with a speed of 5.0 $\mathrm{km} / \mathrm{h}$ parallel to its banks. The hunter uses a small powerboat that moves at a maximum speed of 12 $\mathrm{km} / \mathrm{h}$ with respect to the water. What is the minimum time necessary for crossing?

Averell H.
Carnegie Mellon University

Problem 50

Chinook salmon are able to move upstream faster by jumping out of the water periodically; this behavior is called porpoising. Suppose a salmon swimming in still water jumps out of the water with a speed of 6.26 $\mathrm{m} / \mathrm{s}$ at an angle of $45^{\circ},$ sails through the air a distance $L$ before returning to the water, and then swims a distance $L$ underwater at a speed of 3.58 $\mathrm{m} / \mathrm{s}$ before beginning another porpoising maneuver. Determine the average speed of the fish.

Averell H.
Carnegie Mellon University

Problem 51

A daredevil is shot out of a cannon at $45.0^{\circ}$ to the horizontal
with an initial speed of $25.0 \mathrm{~m} / \mathrm{s}$. A net is positioned a horizontal distance of $50.0 \mathrm{~m}$ from the cannon. At what height above the cannon should the net be placed in order to catch the claredevil?

Averell H.
Carnegie Mellon University

Problem 52

If raindrops are falling vertically at 7.50 $\mathrm{m} / \mathrm{s}$ , what angle from the vertical do they make for a person jogging at 2.25 $\mathrm{m} / \mathrm{s?}$

Averell H.
Carnegie Mellon University

Problem 53

A celebrated Mark Twain story has motivated contestants in the Calaveras County Jumping Frog Jubilee, where frog jumps as long as 2.2 $\mathrm{m}$ have been recorded. If a frog jumps 2.2 $\mathrm{m}$ and the launch angle is $45^{\circ},$ find $(\mathrm{a})$ the frog's launch speed and (b) the time the frog spends in the air. Ignore air resistance.

Averell H.
Carnegie Mellon University

Problem 54

A landscape architect is planning an artificial waterfall in a city park. Water flowing at 0.750 $\mathrm{m} / \mathrm{s}$ leaves the end of a horizontal channel at the top of a vertical wall $h=2.35 \mathrm{m}$ high and falls into a pool (Fig. P3.54). (a) How far from the wall will the water land? Will the space behind the waterfall be wide enough for a pedesbe wide enough for a pedestrian walkway? (b) To sell her plan to the city council, the architect wants to build a model to standard scale, one-twelfth actual size. How fast should the water flow in the channel in the model?

Averell H.
Carnegie Mellon University

Problem 55

A golf ball with an initial speed of 50.0 $\mathrm{m} / \mathrm{s}$ lands exactly 240 $\mathrm{m}$ downrange on a level course. (a) Neglecting air friction, what two projection angles would achieve this result? (b) What is the maximum height reached by the ball, using the two angles determined in part (a)?

Averell H.
Carnegie Mellon University

Problem 56

Antlion larvae lie in wait for prey at the bottom of a conical pit about 5.0 $\mathrm{cm}$ deep and 3.8 $\mathrm{cm}$ in radius. When a small insect ventures into the pit, it slides to the bottom and is seized by the antlion. If the prey attempts to escape, the antlion rapidly launches grains of sand at the prey, cither knocking it down or causing a small avalanche that returns the prey to the bottom of the pit. Suppose an antlion launches grains of sand at an angle of $72^{\circ}$ above the horizon. Find the launch speed $v_{0}$ required to hit a target at the top of the pit, 5.0 $\mathrm{cm}$ above and 3.8 $\mathrm{cm}$ to the right of the antlion.

Averell H.
Carnegie Mellon University

Problem 57

One strategy in a snowball fight is to throw a snowball at a high angle over level ground. Then, while your opponent is watching that snowball, you throw a second one at a low angle timed to arrive before or at the same time as the first one. Assume both snowballs are thrown with a speed of 25.0 $\mathrm{m} / \mathrm{s}$ . The first is thrown at an angle of $70.0^{\circ}$ with respect to the horizontal. (a) At what angle should the second snowball be thrown to arrive at the same point as the first? (b) How many seconds later should the second snowball be thrown after the first for both to arrive at the same time?

Averell H.
Carnegie Mellon University

Problem 58

A football receiver running straight downfield at 5.50 $\mathrm{m} / \mathrm{s}$ is 10.0 $\mathrm{m}$ in front of the quarterback when a pass is thrown downfield at $25.0^{\circ}$ above the horizon (Fig. P3.58). If the receiver never changes speed and the ball is caught at the same height from which it was thrown, find (a) the foothall's initial speed, (b) the amount of time the football spends in the air, and (c) the distance between the quarterback and the receiver when the catch is made.

Averell H.
Carnegie Mellon University

Problem 59

The determined wile E. Coyote is out once more to try to capture the elusive roadrunner. The coyote wears a new pair of power roller skates, which provide a constant horizontal acceleration of $15.0 \mathrm{m} / \mathrm{s}^{2},$ as shown in Figure $\mathrm{P} 3.59$ . The coyote starts off at rest 70.0 $\mathrm{m}$ from the edge of a cliff at the instant the road-runner zips by in the direction of the cliff. (a) If the roadrunner moves with constant speed, find the minimum speed the roadrunner must have to reach the cliff before the coyote. (b) If the cliff is $1.00 \times 10^{2} \mathrm{m}$ above the base of a canyon, find where the coyote lands in the canyon. (Assume his skates are still in operation when he is in "flight" and that his horizontal component of acceleration remains constant at 15.0 $\mathrm{m} / \mathrm{s}^{2}$ )

Averell H.
Carnegie Mellon University