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Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett

Chapter 2

Motion in One Dimension - all with Video Answers

Educators

Chapter Questions

02:31

Problem 1

The position versus time for a certain particle moving along the $x$ axis is shown in Figure $\mathrm{P} 2.1 .$ Find the average velocity in the following time intervals. (a) 0 to $2 \mathrm{~s} \quad$ (b) 0
$\begin{array}{lllll}\text { to } 4 \mathrm{~s} & \text { (c) } 2 \mathrm{~s} \text { to } 4 \mathrm{~s} & \text { (d) } 4 \mathrm{~s} \text { to } 7 \mathrm{~s} & \text { (e) } 0 \text { to } 8 \mathrm{~s}\end{array}$
Figure P2.1 Problems 1 and 8 .

Efren Serra
Efren Serra
Numerade Educator
02:01

Problem 2

The position of a pinewood derby car was observed at various moments; the results are summarized in the following table. Find the average velocity of the car for (a) the first 1-s time interval, (b) the last $3 \mathrm{~s}$, and (c) the entire period of observation. \begin{tabular}{llllrrr}
\hline$t(\mathbf{s})$ & 0 & $1.0$ & $2.0$ & $3.0$ & $4.0$ & $5.0$ \\
\hline$x(\mathbf{m})$ & 0 & $2.3$ & $9.2$ & $20.7$ & $36.8$ & $57.5$ \\
\hline
\end{tabular}

Keshav Singh
Keshav Singh
Numerade Educator
03:23

Problem 3

A person walks first at a constant speed of $5.00 \mathrm{~m} / \mathrm{s}$ along a straight line from point $A$ to point $B$ and then back along the line from $B$ to $A$ at a constant speed of $3.00 \mathrm{~m} / \mathrm{s}$. (a) What is her average speed over the entire trip?
(b) What is her average velocity over the entire trip?

Keshav Singh
Keshav Singh
Numerade Educator
03:22

Problem 4

A particle moves according to the equation $x=10 t^{2}$, where $x$ is in meters and $t$ is in seconds. (a) Find the average velocity for the time interval from $2.00 \mathrm{~s}$ to $3.00 \mathrm{~s}$.
(b) Find the average velocity for the time interval from $2.00 \mathrm{~s}$ to $2.10 \mathrm{~s}$

Andrew Contreras
Andrew Contreras
Numerade Educator
02:41

Problem 5

A A position-time graph for a particle moving along the $x$ axis is shown in Figure P2.5. (a) Find the average velocity in the time interval $t=1.50 \mathrm{~s}$ to $t=4.00 \mathrm{~s}$. (b) Determine the instantaneous velocity at $t=2.00 \mathrm{~s}$ by measuring the slope of the tangent line shown in the graph. (c) At what value of $t$ is the velocity zero?
Fiqure $\mathrm{P} 2.5$

Efren Serra
Efren Serra
Numerade Educator
04:06

Problem 6

The position of a particle moving along the $x$ axis varies in time according to the expression $x=3 t^{2}$, where $x$ is in meters and $t$ is in seconds. Evaluate its position (a) at $t=$ $3.00 \mathrm{~s}$ and $(\mathrm{b})$ at $3.00 \mathrm{~s}+\Delta t .(\mathrm{c})$ Evaluate the limit of $\Delta x / \Delta t$ as $\Delta t$ approaches zero to find the velocity at $t=$ $3.00 \mathrm{~s} .$

Andrew Contreras
Andrew Contreras
Numerade Educator
03:32

Problem 7

(a) Use the data in Problem $2.2$ to construct a smooth graph of position versus time. (b) By constructing tangents to the $x(t)$ curve, find the instantaneous velocity of the car at several instants. (c) Plot the instantaneous velocity versus time and, from the graph, determine the average acceleration of the car. (d) What was the initial

Keshav Singh
Keshav Singh
Numerade Educator
02:26

Problem 8

Find the instantaneous velocity of the particle described in Figure $\mathrm{P} 2.1$ at the following times: (a) $t=1.0 \mathrm{~s}$ (b) $t=$ $3.0 \mathrm{~s}(\mathrm{c}) t=4.5 \mathrm{~s}(\mathrm{~d}) t=7.5 \mathrm{~s}$

Mayukh Banik
Mayukh Banik
Numerade Educator
01:39

Problem 9

A hare and a tortoise compete in a race over a course $1.00 \mathrm{~km}$ long. The tortoise crawls straight and steadily at its maximum speed of $0.200 \mathrm{~m} / \mathrm{s}$ toward the finish line. The hare runs at its maximum speed of $8.00 \mathrm{~m} / \mathrm{s}$ toward the goal for $0.800 \mathrm{~km}$ and then stops to tease the tortoise. How close to the goal can the hare let the tortoise approach before resuming the race, which the tortoise wins in a photo finish? Assume both animals, when moving, move steadily at their respective maximum speeds.

Keshav Singh
Keshav Singh
Numerade Educator
01:35

Problem 10

A $50.0-\mathrm{g}$ Super Ball traveling at $25.0 \mathrm{~m} / \mathrm{s}$ bounces off a brick wall and rebounds at $22.0 \mathrm{~m} / \mathrm{s}$. A high-speed camera records this event. If the ball is in contact with the wall for $3.50 \mathrm{~ms}$, what is the magnitude of the average acceleration of the ball during this time interval? Note:
$1 \mathrm{~ms}=10^{-3} \mathrm{~s}$

Prashant Bana
Prashant Bana
Numerade Educator
08:11

Problem 11

A particle starts from rest and accelerates as shown in Figure P2.11. Determine (a) the particle's speed at $t=10.0 \mathrm{~s}$ and at $t=20.0 \mathrm{~s}$ and $(\mathrm{b})$ the distance traveled in the first $20.0 \mathrm{~s}$.
Figure P2.11

Nathan Silvano
Nathan Silvano
Numerade Educator
04:44

Problem 12

A velocity-time graph for an object moving along the $x$ axis is shown in Figure P2.12. (a) Plot a graph of the acceleration versus time. (b) Determine the average acceleration of the object in the time intervals $t=5.00 \mathrm{~s}$ to $t=$ $15.0 \mathrm{~s}$ and $t=0$ to $t=20.0 \mathrm{~s}$.

Sachin Rao
Sachin Rao
Numerade Educator
02:18

Problem 13

A particle moves along the $x$ axis a equation $x=2.00+3.00 t-1.00 t^{2}$, where $x$ is in meters and $t$ is in seconds. At $t=3.00 \mathrm{~s}$, find $(\mathrm{a})$ the position of the particle, (b) its velocity, and (c) its acceleration.

Efren Serra
Efren Serra
Numerade Educator
04:45

Problem 14

A child rolls a marble on a bent track that is $100 \mathrm{~cm}$ long as shown in Figure P2.14. We use $x$ to represent the position of the marble along the track. On the horizontal sections from $x=0$ to $x=20 \mathrm{~cm}$ and from $x=40 \mathrm{~cm}$ to $x=$ $60 \mathrm{~cm}$, the marble rolls with constant speed. On the sloping sections, the speed of the marble changes steadily. At the places where the slope changes, the marble stays on the track and does not undergo any sudden changes in speed. The child gives the marble some initial speed at $x=0$ and $t=0$ and then watches it roll to $x=90 \mathrm{~cm}$ where it turns around, eventually returning to $x=0$ with the same speed with which the child initially released it. Prepare graphs of $x$ versus $t, v_{x}$ versus $t$, and $a_{x}$ versus $t$, vertically aligned with their time axes identical, to show the motion of the marble. You will not be able to place numbers other than zero on the horizontal axis or on the velocity or acceleration axes, but show the correct relative sizes on the graphs.
Figure P2.14

Mayukh Banik
Mayukh Banik
Numerade Educator
04:15

Problem 15

An object moves along the $x$ axis according to the equation $x(t)=\left(3.00 t^{2}-2.00 t+3.00\right) \mathrm{m}$, where $t$ is in $\sec$ onds. Determine (a) the average speed between $t=2.00 \mathrm{~s}$ and $t=3.00 \mathrm{~s}$, (b) the instantaneous speed at $t=2.00 \mathrm{~s}$ and at $t=3.00 \mathrm{~s},(\mathrm{c})$ the average acceleration between $t=2.00 \mathrm{~s}$ and $t=3.00 \mathrm{~s}$, and $(\mathrm{d})$ the instantaneous acceleration at $t=2.00 \mathrm{~s}$ and $t=3.00 \mathrm{~s}$.

Sachin Rao
Sachin Rao
Numerade Educator
03:07

Problem 16

Figure $\mathrm{P} 2.16$ shows a graph of $v_{x}$ versus $t$ for the motion of a motorcyclist as he starts from rest and moves along Figure P2.16
the road in a straight line. (a) Find the average acceleration for the time interval $t=0$ to $t=6.00 \mathrm{~s}$. (b) Estimate the time at which the acceleration has its greatest positive value and the value of the acceleration at that instant.
(c) When is the acceleration zero?
(d) Estimate the maximum negative value of the acceleration and the time at which it occurs.

Keshav Singh
Keshav Singh
Numerade Educator
02:47

Problem 17

Each of the strobe photographs (a), (b), and (c) in Figure Q2.7 was taken of a single disk moving toward the right, which we take as the positive direction. Within each photograph the time interval between images is constant. For each photograph, prepare graphs of $x$ versus $t, v_{x}$ versus $t$, and $a_{x}$ versus $t$, vertically aligned with their time axes identical, to show the motion of the disk. You will not be able to place numbers other than zero on the axes, but show the correct relative sizes on the graphs.

Keshav Singh
Keshav Singh
Numerade Educator
01:30

Problem 18

Draw motion diagrams for (a) an object moving to the right at constant speed, (b) an object moving to the right and speeding up at a constant rate, (c) an object moving to the right and slowing down at a constant rate,
(d) an object moving to the left and speeding up at a constant rate, and (e) an object moving to the left and slowing down at a constant rate. (f) How would your drawings change if the changes in speed were not uniform; that is, if the speed were not changing at a constant rate?

Mayukh Banik
Mayukh Banik
Numerade Educator
04:32

Problem 19

Assume a parcel of air in a straight tube moves with a constant acceleration of $-4.00 \mathrm{~m} / \mathrm{s}^{2}$ and has a velocity of $13.0 \mathrm{~m} / \mathrm{s}$ at $10: 05: 00 \mathrm{a} . \mathrm{m} .$ on a certain date.
(a) What is its $\begin{array}{llllll}\text { velocity at } & 10: 05: 01 & \text { a.m.? } & \text { (b) At } 10: 05: 02 & \text { a.m.? } & \text { (c) } \mathrm{At}\end{array}$ 10:05:02.5 a.m.?
(d) At $10: 05: 04$ a.m.?
(e) At $10: 04: 59$
a.m.? (f) Describe the shape of a graph of velocity versus time for this parcel of air. (g) Argue for or against the statement, "Knowing the single value of an object's constant acceleration is like knowing a whole list of values for its velocity."

Keshav Singh
Keshav Singh
Numerade Educator
01:54

Problem 20

A truck covers $40.0 \mathrm{~m}$ in $8.50 \mathrm{~s}$ while smoothly slowing down to a final speed of $2.80 \mathrm{~m} / \mathrm{s}$. (a) Find its original speed. (b) Find its acceleration.

Mayukh Banik
Mayukh Banik
Numerade Educator
01:42

Problem 21

A An object moving with uniform acceleration has a velocity of $12.0 \mathrm{~cm} / \mathrm{s}$ in the positive $x$ direction when its $x$ coordinate is $3.00 \mathrm{~cm}$. If its $x$ coordinate $2.00 \mathrm{~s}$ later is $-5.00 \mathrm{~cm}$, what is its acceleration?

Efren Serra
Efren Serra
Numerade Educator
09:19

Problem 22

Figure P2.22 represents part of the performance data of a car owned by a proud physics student. (a) Calculate the total distance traveled by computing the area under the graph line. (b) What distance does the car travel between the times $t=10 \mathrm{~s}$ and $t=40 \mathrm{~s} ?$ (c) Draw a graph of its acceleration versus time between $t=0$ and $t=50 \mathrm{~s}$.
(d) Write an equation for $x$ as a function of time for each phase of the motion, represented by (i) $0 a$, (ii) $a b$, and
(iii) $b c .$ (e) What is the average velocity of the car between $t=0$ and $t=50 \mathrm{~s}$ ?
Figure P2.22

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:37

Problem 23

A jet plane comes in for a landing with a speed of $100 \mathrm{~m} / \mathrm{s}$, and its acceleration can have a maximum magnitude of $5.00 \mathrm{~m} / \mathrm{s}^{2}$ as it comes to rest. (a) From the instant the plane touches the runway, what is the minimum time interval needed before it can come to rest?
(b) Can this plane land on a small tropical island airport where the runway is $0.800 \mathrm{~km}$ long? Explain your answer.

Keshav Singh
Keshav Singh
Numerade Educator
05:54

Problem 24

At $t=0$, one toy car is set rolling on a straight track with initial position $15.0 \mathrm{~cm}$, initial velocity $-3.50 \mathrm{~cm} / \mathrm{s}$ and constant acceleration $2.40 \mathrm{~cm} / \mathrm{s}^{2} .$ At the same moment, another toy car is set rolling on an adjacent track with initial position $10.0 \mathrm{~cm}$, an initial velocity of $+5.50 \mathrm{~cm} / \mathrm{s}$, and constant acceleration zero. (a) At what time, if any, do the two cars have equal speeds? (b) What are their speeds at that time? (c) At what time(s), if any, do the cars pass each other? (d) What are their locations at that time? (e) Explain the difference between question
(a) and question (c) as clearly as possible. Write (or draw) for a target audience of students who do not immediately understand the conditions are different.

Massimo Antonelli
Massimo Antonelli
Numerade Educator
03:08

Problem 25

The driver of a car slams on the brakes when he sees a tree blocking the road. The car slows uniformly with an acceleration of $-5.60 \mathrm{~m} / \mathrm{s}^{2}$ for $4.20 \mathrm{~s}$, making straight skid marks $62.4 \mathrm{~m}$ long ending at the tree. With what speed does the car then strike the tree?

Keshav Singh
Keshav Singh
Numerade Educator
04:03

Problem 26

Help! One of our equations is missing! We describe constantacceleration motion with the variables and parameters $v_{x i}$ $v_{x f} a_{x}, t$, and $x_{f}-x_{i}$, Of the equations in Table $2.2$, the first does not involve $x_{f}-x_{i}$, the second does not contain $a_{x}$, the third omits $v_{x f}$ and the last leaves out $t .$ So, to complete the set there should be an equation not involving $v_{x i}$ Derive it from the others. Use it to solve Problem 25 in
one step.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:46

Problem 27

For many years Colonel John P. Stapp, USAF, held the world's land speed record. He participated in studying whether a jet pilot could survive emergency ejection. On March 19,1954 , he rode a rocket-propelled sled that moved down a track at a speed of $632 \mathrm{mi} / \mathrm{h} .$ He and the

Vishal Gupta
Vishal Gupta
Numerade Educator
02:56

Problem 28

A particle moves along the $x$ axis. Its position is given by the equation $x=2+3 t-4 t^{2}$, with $x$ in meters and $t$ in seconds. Determine (a) its position when it changes direction and (b) its velocity when it returns to the position it had at $t=0$.

Mayukh Banik
Mayukh Banik
Numerade Educator
03:47

Problem 29

An electron in a cathode-ray tube accelerates from a speed of $2.00 \times 10^{4} \mathrm{~m} / \mathrm{s}$ to $6.00 \times 10^{6} \mathrm{~m} / \mathrm{s}$ over $1.50 \mathrm{~cm}$
(a) In what time interval does the electron travel this $1.50 \mathrm{~cm} ?$ (b) What is its acceleration?

Keshav Singh
Keshav Singh
Numerade Educator
09:31

Problem 30

Within a complex machine such as a robotic assembly line, suppose one particular part glides along a straight track. A control system measures the average velocity of the part during each successive time interval $\Delta t_{0}=t_{0}-0$, compares it with the value $v_{c}$ it should be, and switches a servo motor on and off to give the part a correcting pulse of acceleration. The pulse consists of a constant acceleration $a_{m}$ applied for time interval $\Delta t_{m}=t_{m}-0$ within the next control time interval $\Delta t_{0}$. As shown in Figure $\mathrm{P} 2.30$, the part may be modeled as having zero acceleration when the motor is off (between $t_{m}$ and $t_{0}$ ). A computer in the control system chooses the size of the acceleration so that the final velocity of the part will have the correct value $v_{e}$. Assume the part is initially at rest and is to have instantaneous velocity $v_{c}$ at time $t_{0}$. (a) Find the required value of $a_{m}$ in terms of $v_{c}$ and $t_{m}$
(b) Show that the displacement $\Delta x$ of the part during the time interval $\Delta t_{0}$ is given by $\Delta x=v_{c}\left(t_{0}-0.5 t_{m}\right) .$ For specified values of $v_{c}$ and $t_{0},(\mathrm{c})$ what is the minimum displacement of the part?
(d) What is the maximum displacement of the part?
(e) Are both the minimum and maximum displacements physically attainable?
Figure

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:41

Problem 31

A glider on an air track carries a flag of length $\ell$ through a stationary photogate, which measures the time interval $\Delta t_{d}$ during which the flag blocks a beam of infrared light passing across the photogate. The ratio $v_{d}=$ $\ell / \Delta t_{d}$ is the average velocity of the glider over this part of its motion. Suppose the glider moves with constant acceleration. (a) Argue for or against the idea that $v_{d}$ is equal to the instantaneous velocity of the glider when it is halfway through the photogate in space. (b) Argue for or against the idea that $v_{d}$ is equal to the instantaneous velocity of the glider when it is halfway through the photogate in time.

Keshav Singh
Keshav Singh
Numerade Educator
05:44

Problem 32

Speedy Sue, driving at $30.0 \mathrm{~m} / \mathrm{s}$, enters a one-lane tunnel. She then observes a slow-moving van $155 \mathrm{~m}$ ahead traveling at $5.00 \mathrm{~m} / \mathrm{s}$. Sue applies her brakes but can accelerate only at $-2.00 \mathrm{~m} / \mathrm{s}^{2}$ because the road is wet. Will there be a collision? State how you decide. If yes, determine how far into the tunnel and at what time the collision occurs. If no, determine the distance of closest approach between Sue's car and the van.

Keshav Singh
Keshav Singh
Numerade Educator
05:30

Problem 33

Vroom, vroom! As soon as a traffic light turns green, a car speeds up from rest to $50.0 \mathrm{mi} / \mathrm{h}$ with constant acceleration $9.00 \mathrm{mi} / \mathrm{h} \cdot \mathrm{s}$. In the adjoining bike lane, a cyclist speeds up from rest to $20.0 \mathrm{mi} / \mathrm{h}$ with constant acceleration $13.0 \mathrm{mi} / \mathrm{h} \cdot \mathrm{s}$. Each vehicle maintains constant velocity after reaching its cruising speed.
(a) For what time interval is the bicycle ahead of the car? (b) By what maximum distance does the bicycle lead the car?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:21

Problem 34

Solve Example $2.8$ (Watch Out for the Speed Limit!) by a graphical method. On the same graph plot position versus time for the car and the police officer. From the intersection of the two curves read the time at which the trooper overtakes the car.

Keshav Singh
Keshav Singh
Numerade Educator
View

Problem 35

A glider of length $12.4 \mathrm{~cm}$ moves on an air track with constant acceleration. A time interval of $0.628 \mathrm{~s}$ elapses between the moment when its front end passes a fixed point $(1)$ along the track and the moment when its back end passes this point. Next, a time interval of $1.39 \mathrm{~s}$ elapses between the moment when the back end of the glider passes point $@$ and the moment when the front end of the glider passes a second point (B) farther down the track. After that, an additional $0.431$ s elapses until the back end of the glider passes point (B). (a) Find the average speed of the glider as it passes point $($ A. (b) Find the acceleration of the glider. (c) Explain how you can compute the acceleration without knowing the distance between points $\mathbb{A}$ and (B).

Gregory Devenport
Gregory Devenport
Numerade Educator
05:19

Problem 36

In a classic clip on America's Funniest Home Videos, a sleeping cat rolls gently off the top of a warm TV set. Ignoring air resistance, calculate (a) the position and (b) the velocity of the cat after $0.100 \mathrm{~s}, 0.200 \mathrm{~s}$, and $0.300 \mathrm{~s}$.

Keshav Singh
Keshav Singh
Numerade Educator
06:08

Problem 37

Every morning at seven o'clock There's twenty terriers drilling on the rock. The boss comes around and he says, "Keep still And bear down heavy on the cast-iron drill
And drill, ye terriers, drill." And drill, ye terriers, drill. It's work all day for sugar in your tea Down beyond the railway. And drill, ye terriers, drill.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:21

Problem 38

A ball is thrown directly downward, with an initial speed of $8.00 \mathrm{~m} / \mathrm{s}$, from a height of $30.0 \mathrm{~m}$. After what time interval does the ball strike the ground?

Bret Rosen
Bret Rosen
Numerade Educator
02:38

Problem 39

A A student throws a set of keys vertically upward to her sorority sister, who is in a window $4.00 \mathrm{~m}$ above. The keys are caught $1.50 \mathrm{~s}$ later by the sister's outstretched hand.
(a) With what initial velocity were the keys thrown?
(b) What was the velocity of the keys just before they were caught?

Keshav Singh
Keshav Singh
Numerade Educator
02:00

Problem 40

Emily challenges her friend David to catch a dollar bill is follows. She holds the bill vertically, as shown in Figure $2.40$, with the center of the bill between David's index inger and thumb. David must catch the bill after Emily
-eleases it without moving his hand downward. If his reacion time is $0.2 \mathrm{~s}$, will he succeed? Explain your reasoning.
Figure P2.40

Keshav Singh
Keshav Singh
Numerade Educator
02:16

Problem 41

A baseball is hit so that it travels straight upward after being struck by the bat. A fan observes that it takes $3.00 \mathrm{~s}$ for the ball to reach its maximum height. Find (a) the ball's initial velocity and (b) the height it reaches.

Keshav Singh
Keshav Singh
Numerade Educator
05:16

Problem 42

An attacker at the base of a castle wall $3.65 \mathrm{~m}$ high throws a rock straight up with speed $7.40 \mathrm{~m} / \mathrm{s}$ at a height of $1.55 \mathrm{~m}$ above the ground. (a) Will the rock reach the top of the wall? (b) If so, what is its speed at the top? If not, what initial speed must it have to reach the top? (c) Find the change in speed of a rock thrown straight down from the top of the wall at an initial speed of $7.40 \mathrm{~m} / \mathrm{s}$ and moving between the same two points.
(d) Does the change in speed of the downward-moving rock agree with the magnitude of the speed change of the rock moving upward between the same elevations? Explain physically why it does or does not agree.

Keshav Singh
Keshav Singh
Numerade Educator
01:12

Problem 43

a drop vertically onto a horse galloping under the tree. The constant speed of the horse is $10.0 \mathrm{~m} / \mathrm{s}$, and the distance from the limb to the level of the saddle is $3.00 \mathrm{~m}$. (a) What must the horizontal distance between the saddle and limb be when the ranch hand makes his move? (b) For what time interval is he in the air?

Efren Serra
Efren Serra
Numerade Educator
01:40

Problem 44

The height of a helicopter above the ground is given by $h$ $=3.00 t^{3}$, where $h$ is in meters and $t$ is in seconds. After $2.00 \mathrm{~s}$, the helicopter releases a small mailbag. How long after its release does the mailbag reach the ground?

Anand Jangid
Anand Jangid
Numerade Educator
03:47

Problem 45

A freely falling object requires $1.50 \mathrm{~s}$ to travel the last $30.0 \mathrm{~m}$ before it hits the ground. From what height above the ground did it fall?

Keshav Singh
Keshav Singh
Numerade Educator
05:42

Problem 46

A student drives a moped along a straight road as described by the velocity-versus-time graph in Figure P2.46. Sketch this graph in the middle of a sheet of graph paper. (a) Directly above your graph, sketch a graph of the position versus time, aligning the time coordinates of the two graphs. (b) Sketch a graph of the acceleration versus time directly below the $v_{x}-t$ graph, again aligning the time coordinates. On each graph, show the numerical values of $x$ and $a_{x}$ for all points of inflection. (c) What is the acceleration at $t=6 \mathrm{~s} ?(\mathrm{~d})$ Find the position (relative to the starting point) at $t=6 \mathrm{~s}$. (e) What is the moped's final position at $t=9 \mathrm{~s}$ ?
Figure P2.46

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:32

Problem 47

Automotive engineers refer to the time rate of change of acceleration as the "jerk." Assume an object moves in one dimension such that its jerk $J$ is constant. (a) Determine expressions for its acceleration $a_{x}(t)$, velocity $v_{x}(t)$, and position $x(t)$, given that its initial acceleration, velocity, and position are $a_{x i} v_{x i}$, and $x_{i}$, respectively. (b) Show that $a_{x}^{2}=a_{x i}^{2}+2 J\left(v_{x}-v_{x x}\right)$

Efren Serra
Efren Serra
Numerade Educator
07:12

Problem 48

The speed of a bullet as it travels down the barrel of a rifle toward the opening is given by $v=\left(-5.00 \times 10^{7}\right) t^{2}+$ $\left(3.00 \times 10^{5}\right) t$, where $v$ is in meters per second and $t$ is in seconds. The acceleration of the bullet just as it leaves the barrel is zero. (a) Determine the acceleration and position of the bullet as a function of time when the bullet is in the barrel. (b) Determine the time interval over which the bullet is accelerated. (c) Find the speed at which the bullet leaves the barrel. (d) What is the length of the barrel?

Keshav Singh
Keshav Singh
Numerade Educator
07:55

Problem 49

An object is at $x=0$ at $t=0$ and moves along the $x$ axis according to the velocity-time graph in Figure $\mathrm{P} 2.49 .$
(a) What is the acceleration of the object between 0 and 4 s?
(b) What is the acceleration of the object between $4 \mathrm{~s}$ and 9 s? (c) What is the acceleration of the object between $13 \mathrm{~s}$ and $18 \mathrm{~s}$ ? $(\mathrm{d})$ At what time(s) is the object moving with the lowest speed? (e) At what time is the object farthest from $x=0 ?$ (f) What is the final position $x$ of the object at $t=18 \mathrm{~s} ?$ (g) Through what total distance has the object moved between $t=0$ and $t=18 \mathrm{~s}$ ?
Figure $\mathrm{P} 2.49$

Sheh Lit Chang
Sheh Lit Chang
University of Washington
09:26

Problem 50

The Acela (pronounced ah-SELL-ah and shown in Fig. P2.50a) is an electric train on the Washington-New York-Boston run, carrying passengers at $170 \mathrm{mi} / \mathrm{h}$. The carriages tilt as much as $6^{\circ}$ from the vertical to prevent passengers from feeling pushed to the side as they go around curves. A velocity-time graph for the Acela is shown in Figure P2.50b. (a) Describe the motion of the train in each successive time interval. (b) Find the peak positive acceleration of the train in the motion graphed.
(c) Find the train's displacement in miles between $t=0$ and $t=200 \mathrm{~s}$
(a)
(b)
Figure P2.50 (a) The Acela: 1171000 tb of cold steel thundering along with 304 passengers. (b) Velocity-versus-time graph for the Acela.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:47

Problem 51

A test rocket is fired vertically upward from a well. A catapult gives it an initial speed of $80.0 \mathrm{~m} / \mathrm{s}$ at ground level. Its engines then fire and it accelerates upward at $4.00 \mathrm{~m} / \mathrm{s}^{2}$ until it reaches an altitude of $1000 \mathrm{~m}$. At that point its engines fail and the rocket goes into free fall, with an acceleration of $-9.80 \mathrm{~m} / \mathrm{s}^{2}$. (a) For what time interval is the rocket in motion above the ground?
(b) What is its maximum altitude? (c) What is its velocity just before it collides with the Earth? (You will need to consider the motion while the engine is operating separate from the free-fall motion.)

Prabhu Ramji
Prabhu Ramji
Numerade Educator
02:03

Problem 52

In Active Figure 2.11b, the area under the velocity versus time curve and between the vertical axis and time $t$ (vertical dashed line) represents the displacement. As shown, this area consists of a rectangle and a triangle. Compute their areas and state how the sum of the two areas compares with the expression on the right-hand side of Equation $2.16$.

Supratim Pal
Supratim Pal
Numerade Educator
09:59

Problem 53

Setting a world record in a 100 -m race, Maggie and Judy cross the finish line in a dead heat, both taking $10.2 \mathrm{~s}$. Accelerating uniformly, Maggie took $2.00 \mathrm{~s}$ and Judy took $3.00 \mathrm{~s}$ to attain maximum speed, which they maintained for the rest of the race. (a) What was the acceleration of each sprinter? (b) What were their respective maximum speeds? (c) Which sprinter was ahead at the $6.00$ -s mark, and by how much?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
08:25

Problem 54

How long should a traffic light stay yellow? Assume you are driving at the speed limit $v_{0}$. As you approach an intersection $22.0 \mathrm{~m}$ wide, you see the light turn yellow. During your reaction time of $0.600 \mathrm{~s}$, you travel at constant speed as you recognize the warning, decide whether to stop or to go through the intersection, and move your foot to the brake if you must stop. Your car has good brakes and can accelerate at $-2.40 \mathrm{~m} / \mathrm{s}^{2} .$ Before it turns red, the light should stay yellow long enough for you to be able to get to the other side of the intersection without speeding up, if you are too close to the intersection to stop before entering it. (a) Find the required time interval $\Delta t_{y}$ that the light should stay yellow in terms of $v_{0} .$ Evaluate your answer for (b) $v_{0}=8.00 \mathrm{~m} / \mathrm{s}=28.8 \mathrm{~km} / \mathrm{h}$, (c) $v_{0}=$
$11.0 \mathrm{~m} / \mathrm{s}=40.2 \mathrm{~km} / \mathrm{h}$, (d) $v_{0}=18.0 \mathrm{~m} / \mathrm{s}=64.8 \mathrm{~km} / \mathrm{h}$
and $(\mathrm{e}) v_{0}=25.0 \mathrm{~m} / \mathrm{s}=90.0 \mathrm{~km} / \mathrm{h} .$ What If? Evaluate
your answer for (f) $v_{0}$ approaching zero, and (g) $v_{0}$ approaching infinity. (h) Describe the pattern of variation of $\Delta t_{y}$ with $v_{0}$. You may wish also to sketch a graph of it. Account for the answers to parts (f) and (g) physically. (i) For what value of $v_{0}$ would $\Delta t_{y}$ be minimal, and (j) what is this minimum time interval? Suggestion:
You may find it easier to do part (a) after first doing part (b).

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:50

Problem 55

A commuter train travels between two downtown stations. Because the stations are only $1.00 \mathrm{~km}$ apart, the train never reaches its maximum possible cruising speed. During rush hour the engineer minimizes the time interval $\Delta t$ between two stations by accelerating for a time interval $\Delta t_{1}$ at a rate $a_{1}=0.100 \mathrm{~m} / \mathrm{s}^{2}$ and then immediately braking with acceleration $a_{2}=-0.500 \mathrm{~m} / \mathrm{s}^{2}$ for a time interval $\Delta t_{2} .$ Find the minimum time interval of travel $\Delta t$ and the time interval $\Delta t_{1}$.

Keshav Singh
Keshav Singh
Numerade Educator
08:05

Problem 56

A Ferrari $F 50$ of length $4.52 \mathrm{~m}$ is moving north on a roadway that intersects another perpendicular roadway. The width of the intersection from near edge to far edge is $28.0 \mathrm{~m}$. The Ferrari has a constant acceleration of magnitude $2.10 \mathrm{~m} / \mathrm{s}^{2}$ directed south. The time interval required for the nose of the Ferrari to move from the near (south) edge of the intersection to the north edge of the intersection is $3.10$ s. (a) How far is the nose of the Ferrari from the south edge of the intersection when it stops? (b) For what time interval is any part of the Ferrari within the boundaries of the intersection? (c) A Corvette is at rest on the perpendicular intersecting roadway. As the nose of the Ferrari enters the intersection, the Corvette starts from rest and accelerates east at $5.60 \mathrm{~m} / \mathrm{s}^{2}$. What is the minimum distance from the near (west) edge of the intersection at which the nose of the Corvette can begin its motion if the Corvette is to enter the intersection after the Ferrari has entirely left the intersection? (d) If the Corvette begins its motion at the position given by your answer to part (c), with what speed does it enter the intersection?

Samuel Smith
Samuel Smith
Numerade Educator
06:10

Problem 57

An inquisitive physics student and mountain climber climbs a $50.0-\mathrm{m}$ cliff that overhangs a calm pool of water. He throws two stones vertically downward, $1.00 \mathrm{~s}$ apart, and observes that they cause a single splash. The first stone has an initial speed of $2.00 \mathrm{~m} / \mathrm{s}$. (a) How long after release of the first stone do the two stones hit the water? (b) What initial velocity must the second stone have if they are to hit simultaneously? (c) What is the speed of each stone at the instant the two hit the water?

Keshav Singh
Keshav Singh
Numerade Educator
04:00

Problem 58

A hard rubber ball, released at chest height, falls to the pavement and bounces back to nearly the same height. When it is in contact with the pavement, the lower side of the ball is temporarily flattened. Suppose the maximum depth of the dent is on the order of $1 \mathrm{~cm}$. Compute an order-of-magnitude estimate for the maximum acceleration of the ball while it is in contact with the pavement. State your assumptions, the quantities you estimate, and the values you estimate for them.

Keshav Singh
Keshav Singh
Numerade Educator
03:59

Problem 59

Kathy Kool buys a sports car that can accelerate at the rate of $4.90 \mathrm{~m} / \mathrm{s}^{2}$. She decides to test the car by racing with another speedster, Stan Speedy. Both start from rest, but experienced Stan leaves the starting line $1.00 \mathrm{~s}$ before Kathy. Stan moves with a constant acceleration of $3.50 \mathrm{~m} / \mathrm{s}^{2}$ and Kathy maintains an acceleration of $4.90 \mathrm{~m} / \mathrm{s}^{2}$. Find (a) the time at which Kathy overtakes Stan, (b) the distance she travels before she catches him, and (c) the speeds of both cars at the instant she overtakes him.

Sachin Rao
Sachin Rao
Numerade Educator
04:42

Problem 60

A rock is dropped from rest into a well. (a) The sound of the splash is heard $2.40 \mathrm{~s}$ after the rock is released from rest. How far below the top of the well is the surface of the water? The speed of sound in air (at the ambient temperature) is $336 \mathrm{~m} / \mathrm{s} .$ (b) What If? If the travel time for the sound is ignored, what percentage error is introduced when the depth of the well is calculated?

Keshav Singh
Keshav Singh
Numerade Educator
01:52

Problem 61

In a California driver's handbook, the following data were given about the minimum distance a typical car travels in stopping from various original speeds. The "thinking distance" represents how far the car travels during the driver's reaction time, after a reason to stop can be seen but before the driver can apply the brakes. The "braking distance" is the displacement of the car after the brakes are applied. (a) Is the thinking-distance data consistent with the assumption that the car travels with constant speed? Explain. (b) Determine the best value of the reaction time suggested by the data. (c) Is the braking-distance data consistent with the assumption that the car travels with constant acceleration? Explain.
(d) Determine the best value for the acceleration suggested by the data.
\begin{tabular}{cccc}
\hline Speed $(\mathrm{mi} / \mathrm{h})$ & Thinking Distance (ft) & Braking Distance (ft) & Total Stopping Distance (ft) \\
\hline 25 & 27 & 34 & 61 \\
35 & 38 & 67 & 105 \\
45 & 49 & 110 & 159 \\
55 & 60 & 165 & 225 \\
65 & 71 & 231 & 302
\end{tabular}

Breanna Ollech
Breanna Ollech
Numerade Educator
07:38

Problem 62

Astronauts on a distant planet toss a rock into the air. With the aid of a camera that takes pictures at a steady rate, they record the height of the rock as a function of time as given in the table in the next column. (a) Find the average velocity of the rock in the time interval between each measurement and the next. (b) Using these average velocities to approximate instantaneous velocities at the midpoints of the time intervals, make a graph of velocity as a function of time. Does the rock move with constant acceleration? If so, plot a straight line of best fit on the graph and calculate its slope to find the acceleration. $$
\begin{array}{cccc}
\hline \text { Time (s) } & \text { Height (m) } & \text { Time (s) } & \text { Height (m) } \\
\hline 0.00 & 5.00 & 2.75 & 7.62 \\
0.25 & 5.75 & 3.00 & 7.25 \\
0.50 & 6.40 & 3.25 & 6.77 \\
0.75 & 6.94 & 3.50 & 6.20 \\
1.00 & 7.38 & 3.75 & 5.52 \\
1.25 & 7.72 & 4.00 & 4.73 \\
1.50 & 7.96 & 4.25 & 3.85 \\
1.75 & 8.10 & 4.50 & 2.86 \\
2.00 & 8.13 & 4.75 & 1.77 \\
2.25 & 8.07 & 5.00 & 0.58 \\
2.50 & 7.90 & &
\end{array}
$$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:26

Problem 63

Two objects, $A$ and $B$, are connected by a rigid rod that has length $L$. The objects slide along perpendicular guide rails as shown in Figure $\mathrm{P} 2.63$. Assume A slides to the left with a constant speed $v$. Find the velocity of $\mathrm{B}$ when $\theta=60.0^{\circ}$.
Figure $\mathrm{P} 2.63$

Keshav Singh
Keshav Singh
Numerade Educator