Let $g(x)=\sqrt{x}$ for $x \geq 0$

a. Find the average rate of change of $g(x)$ with respect to $x$ owerthe intervals $[1,2],[1,1.5]$ and $[1,1+h]$.

b. Make a table of values of the average rate of change of $g$ with respect to $x$ over the interval $[1,1+h]$ for some values of $h$ approaching zero, say $h=0.1,0.01,0.001,0.0001,0.00001$ and $0.000001 .$

c. What does your table indicate is the rate of change of $g(x)$ with respect to $x$ at $x=1 ?$

d. Calculate the limit as $h$ approaches zero of the average rate of change of $g(x)$ with respect to $x$ over the interval $[1,1+h]$ .

## Discussion

## Video Transcript

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## Recommended Questions

The accompanying figure shows the time-todistance graph for a sports car accelerating from a standstill. (FIGURE CAN'T COPY)

a. Estimate the slopes of secant lines $P Q_{1}, P Q_{2}, P Q_{3},$ and $P Q_{4}$ arranging them in order in a table like the one in Figure 2.6 What are the appropriate units for these slopes?

b. Then estimate the car's speed at time $t=20 \mathrm{sec}$

Speed of a car The accompanying figure shows the time-to-distance graph for a sports car accelerating from a standstill.

\begin{equation}

\begin{array}{l}{\text { a. Estimate the slopes of secant lines } P Q_{1}, P Q_{2}, P Q_{3}, \text { and } P Q_{4}} \\ {\text { arranging them in order in a table like the one in Figure } 2.6 .} \\ {\text { What are the appropriate units for these slopes? }} \\ {\text { b. Then estimate the car's speed at time } t=20 \mathrm{sec} \text { . }}\end{array}

\end{equation}

In Exercises 7 and $8,$ a distance-time graph is shown.

(a) Estimate the slopes of the secants $P Q_{1}, P Q_{2}, P Q_{3},$ and $P Q_{4}$ arranging them in order in a table. What is the appropriate unit for these slopes?

(b) Estimate the speed at point $P .$

Accelerating from a Standstill The figure shows the distance-time graph for a 1994 Ford $^{\circ}$ Mustang CobraTM accelerating from a standstill.

The changing velocity of a car is represented in the velocity- versus-time graph shown in Figure P 1.62. (a) Describe everything you can about the motion of the car using the graph. (b) What is the displacement of the car between times 0 s and 45 s? What is the path traveled? (c) What is the average speed of the car during all 70 s? What is the average velocity?

Figure $4-26$ is a position-time graph of the motion of two cars on a road. (Chapter 3 ) a. At what time(s) does one car pass the other? b. Which car is moving faster at $7.0 \mathrm{s} ?$ c. At what time(s) do the cars have the same velocity? d. Over what time interval is car $B$ speeding up all the time? e. Over what time interval is car $B$ slowing down all the time?

The velocity of a car changes over an 8.0 -s time period, as shown in Table $3-6$ a. Plot the velocity-time graph of the motion. b. Determine the displacement of the car during the first $2.0 \mathrm{s}$ c. What displacement does the car have during the first $4.0 \mathrm{s} ?$ d. What is the displacement of the car during the entire $8.0 \mathrm{s} ?$ e. Find the slope of the line between $t=0.0 \mathrm{s}$ and $t=4.0 \mathrm{s} .$ What does this slope represent? f. Find the slope of the line between $t=5.0 \mathrm{s}$ and $t=7.0 \mathrm{s} .$ What does this slope indicate?

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 $t=5.00 \mathrm{s}.$

The changing velocity of a car is represented in the velocity-versus-time graph shown in Figure P 1.61. (a) Describe everything you can about the motion of the car using the graph. (b) What is the displacement of the car between times 10 s and 20 s? (c) What was the average speed of the car?

This question concerns the motion of a car on a straight track; the car's velocity as a function of time is plotted below.

(a) Describe what happened to the car at time $t=1 \mathrm{s}$ .

(b) How does the car's average velocity between time $t=0$ and $t=1 \mathrm{s}$ compare to its average velocity between times $t=1 \mathrm{s}$ and $t=5 \mathrm{s} ?$

(c) What is the displacement of the car from time $t=0$ to time $t=7 \mathrm{s}$ ?

(d) Plot the car's acceleration during this interval as a function of time.

(e) Make a sketch of the object's position during this interval as a function of time. Assume that the car begins at $x=0$ .

(a) In Fig. $2.11,$ what is the instantaneous acceleration of the sports car of Example 2.5 at the time of 14 s from the start? (b) What is the displacement of the car from $t=12.0 \mathrm{s}$ to $t=16.0 \mathrm{s} ?(\mathrm{c})$ What is the average velocity of the car in the 4.0 -s time interval from $12.0 \mathrm{s}$ to $16.0 \mathrm{s} ?$

(FIGURE CANNOT COPY)

Velocity of a car The graph shows the position $s=f(t)$ of a car

thours after 5: 00 P.M. relative to its starting point $s=0$, where $s$

is measured in miles.

a. Describe the velocity of the car. Specifically, when is it speeding up and when is it slowing down?

b. At approximately what time is the car traveling the fastest? The slowest?

(II) A particular automobile can accelerate approximately

as shown in the velocity vs. time graph of Fig, 40 . (The short

flat spots in the curve represent shifting of the gears.) Esti-

mate the average acceleration of the car in $(a)$ second gear;

and $(b)$ fourth gear. (c) What is its average acceleration

through the first four gears?

FIGURE 40 Problem $26 .$ The velocity of a

high-performance automobile as a function of time,

starting from a dead stop. The flat spots in the curve

represent gear shifts.

The figure shows a plot of $v_{x}(t)$ for a car traveling in a straight line. (a) What is $a_{\mathrm{av} . x}$ between $t=6 \mathrm{s}$ and $t=11 \mathrm{s} ?$ (b) What is $v_{\mathrm{av}, x}$ for the same time interval?

(c) What is $v_{\text {av. } x}$ for the interval $t=0$ to $t=20 \mathrm{s} ?$

(d) What is the increase in the car's speed between

$10 \mathrm{s}$ and $15 \mathrm{s} ?$ (e) How far does the car travel from time $t=10 \mathrm{s}$ to time $t=15 \mathrm{s} ?$

(GRAPH CANNOT COPY)

Figure P2.35 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 red-brown 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 the segments $0 a, a b,$ and $b c$ (e) What is the average velocity of the car between $t=0$ and $t=50 \mathrm{s}$ ?

A race car moves such that its position fits the relationship

$$x=(5.0 {m} / {s}) t+\left(0.75 {m} / {s}^{3}\right) t^{3}$$

where x is measured in meters and t in seconds. (a) Plot a graph of the car’s position versus time. (b) Determine the instantaneous velocity of the car at t 5 4.0 s, using time intervals of 0.40 s, 0.20 s, and 0.10 s. (c) Compare the average velocity during the first 4.0 s with the results of part (b).

The velocity graph of an accelerating car is shown.

(a) Use the Midpoint Rule to estimate the average velocity of the car during the first 12 seconds.

(b) At what time was the instantaneous velocity equal to the average velocity?

The velocity graph of an accelerating car is shown. (a) Use the Midpoint rule to estimate the average velocity of the car during the first 12 seconds. (b) At what time was the instantaneous velocity equal to the average velocity?

Two cars, A and B, start side by side and accelerate from rest. The figure shows the graphs of their velocity functions.

(a) Which car is ahead after one minute? Explain.

(b) What is the meaning of the area of the shaded region?

(c) Which car is ahead after two minutes? Explain.

(d) Estimate the time at which the cars are again side by side.

Two cars, $A$ and $B,$ start side by side and accelerate from rest. The figure shows the graphs of their velocity functions.

(a) Which car is ahead after one minute? Explain.

(b) What is the meaning of the area of the shaded region?

(c) Which car is ahead after two minutes? Explain.

(d) Estimate the time at which the cars are again side by side.

Two cars, $A$ and $B,$ start side by side and accelerate from

rest. The figure shows the graphs of their velocity functions.

(a) Which car is ahead after one minute? Explain.

(b) What is the meaning of the area of the shaded region?

(c) Which car is ahead after two minutes? Explain.

(d) Estimate the time at which the cars are again side by side.

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 $A$ ?

Distance from velocity data The accompanying table gives

data for the velocity of a vintage sports car accelerating from 0 to

142 $\mathrm{mi} / \mathrm{h}$ in 36 $\mathrm{sec}(10$ thousandths of an hour).

$$

\begin{array}{l}{\text { a. Use rectangles to estimate how far the car traveled during the }} \\ {36 \text { sec it took to reach } 142 \mathrm{mi} / \mathrm{h} \text { . }} \\ {\text { b. Roughly how many seconds did it take the car to reach the }} \\ {\text { halfway point? About how fast was the car going then? }}\end{array}

$$

For a car moving at $60 \mathrm{mi} / \mathrm{h}$, the equation $d=88 t$ gives the distance in feet $d$ that the car travels in $t$ seconds.

a. Graph the line $d=88 t$.

b. On the same graph you made for part a, graph the line $d=300 .$ What does the intersection of the two lines represent?

c. Use the graph to estimate the number of seconds it takes the car to travel $300 \mathrm{ft}$.

(II) The position of a racing car, which starts from rest at $t=0$ and moves in a straight line, is given as a function of time in the following Table. Estimate $(a)$ its velocity and $(b)$ its acceleration as a function of time. Display each in a

Table and on a graph.

$\begin{array}{lllllll}{t(s)} & {0} & {0.25} & {0.50} & {0.75} & {1.00} & {1.50} & {2.00} & {2.50} \\ {x(m)} & {0} & {0.11} & {0.46} & {1.06} & {1.94} & {4.62} & {8.55} & {13.79} \\ \hline\end{array}$

$\begin{array}{rrrrrrrr}{t(s)} & {3.00} & {3.50} & {4.00} & {4.50} & {5.00} & {5.50} & {6.00} \\ {x(m)} & {20.36} & {28.31} & {37.65} & {48.37} & {60.30} & {73.26} & {87.16}\end{array}$

(III) Two cars approach a street corner at right angles to each other (Fig. 3-47). Car 1 travels at a speed relative to Earth $v_{1\mathrm E} =$ 35 km/h and car 2 at $v_{2\mathrm E} =$ 55 km/h. What is the relative velocity of car 1 as seen by car 2? What is the velocity of car 2 relative to car 1?

Velocity The graph shows the velocity, in feet per second,

of a decelerating car after the driver applies the brakes. Use

the graph to estimate how far the car travels before it comes

to a stop.

The position of a car as a function of time is given by

$x=(50 \mathrm{m})+(-5.0 \mathrm{m} / \mathrm{s}) t+\left(-10 \mathrm{m} / \mathrm{s}^{2}\right) t^{2}$ . (a) What are the ini-

tial position, initial velocity, and acceleration of the car? (b) Plot $x$

versus $t$ for $t=0$ to $t=2.0 \mathrm{s}$ . (c) What distance does the car travel

during the first 1.0 $\mathrm{s} ?$ (d) What is the average velocity of the car

between $t=1.0 \mathrm{s}$ and $t=2.0 \mathrm{s} ?$

Modeling Distance Traveled A car starts from point $P$ at time $t=0$ and travels at 45 mph.

(a)Write an expression $d(t)$ for the distance the car travels from $P$

(b) Graph $y=d(t) .$

(c) What is the slope of the graph in (b)? What does it have to do with the car?

(d) Writing to Learn Create a scenario in which $t$ could have negative values.

(e) Writing to Learn Create a scenario in which the $y$ -intercept of $y=d(t)$ could be $30 .$

[T] The position in feet of a race car along a straight track after $t$ seconds is modeled by the function

$$

s(t)=8 t^{2}-\frac{1}{16} t^{3}

$$

a. Find the average velocity of the vehicle over the following time intervals to four decimal places:

$$

\begin{array}{l}{\text { i. }[4,4.1]} \\ {\text { ii. }[4,4.01]} \\ {\text { iii. }[4,4.001]} \\ {\text { iv. }[4,4,0001]}\end{array}

$$

b. Use a. to draw a conclusion about the instantaneous velocity of the vehicle at $t=4$ seconds.

The minimum stopping distance for a car traveling at a speed of $30 \mathrm{m} / \mathrm{s}$ is $60 \mathrm{m},$ including the distance traveled during the driver's reaction time of $0.50 \mathrm{s}$

a. What is the minimum stopping distance for the same car traveling at a speed of $40 \mathrm{m} / \mathrm{s} ?$

b. Draw a position-versus-time graph for the motion of the car in part a. Assume the car is at $x_{0}=0$ m when the driver first sees the emergency situation ahead that calls for a rapid halt.

Figure 5.9 shows the velocity of a car for $0 \leq t \leq 12$ and the rectangles used to estimate the distance traveled.

(a) Do the rectangles represent a left or a right sum? 3

(b) Do the rectangles lead to an upper or a lower estimate?

(c) What is the value of $n ?$

(d) What is the value of $\Delta t ?$

(e) Give an approximate value for the estimate.

(GRAPH CANT COPY)

The velocity graph of a car accelerating from rest to a speed of 120 km/h over a period of 30 seconds is shown. Estimate the distance traveled during this period.

The position of a car in a drag race is measured each second, and A the results are tabulated below.

$$\begin{array}{|l|c|c|c|c|c|c|}

\hline \text { Time } t(\mathrm{s}) & 0 & 1 & 2 & 3 & 4 & 5 \\

\hline \text { Position } x(\mathrm{m}) & 0 & 1.7 & 6.2 & 17 & 24 & 40 \\

\hline

\end{array}$$

Assuming the acceleration is approximately constant, plot position versus a quantity that should make the graph a straight line. Fit a line to the data, and from it determine the approximate acceleration.

The table shows a speedometer readings at 10-second intervals during a 1-minute period for a car racing at the Daytona International Speedway in Florida.

(a) Estimate the distance the race car traveled during this time period using the velocities at the beginning of the time intervals.

(b) Give another estimate using the velocities at the end of the time periods.

(c) Are your estimates in parts (a) and (b) upper and lower estimates? Explain.

Two cars travel in the same direction along a straight

road. Figure 5.14 shows the velocity, v, of each car at

time t. Car B starts 2 hours after car A and car B reaches

a maximum velocity of 50 km/hr.

(a) For approximately how long does each car travel?

(b) Estimate car A’s maximum velocity.

(c) Approximately how far does each car travel?

(GRAPH CAN'T COPY)

The velocity graph of a car accelerating from rest to a speed of 120 $\mathrm{km} / \mathrm{h}$ over a period of 30 seconds is shown. Estimate the distance traveled during this period.

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

The figure shows the graphs of three functions. One is the position function of a car, one is the velocity of the car, and one is its acceleration. Identify each curve, and explain your choices.

The figure shows the graphs of three functions. One is the position function of a car, one is the velocity of the car, and one is its acceleration. Identify each curve, and explain your choices.

Use the velocity-versus-time graph lines in Figure P 1.55 to determine the change in the position of each car from 0 s to 60 s. Represent the motion of each car mathematically as a function $x(t).$ Their initial positions are A (200 m) and B (-200 m).

The table below shows stopping distances in feet for a car tested 3 times at each of 5 speeds. We hope to create a model that predicts Stopping Distance from the Speed of the car.

$$\begin{array}{l}{\text { Speed (mph)}} & {\text { Stopping Distances (ft) }} \\ {20} & {64,62,59} \\ {30} & {114,118,105} \\ {40} & {153,171,165} \\ {50} & {231,203,238} \\ {60} & {317,321,276}\end{array}$$

a) Explain why a linear model is not appropriate.

b) Re-express the data to straighten the scatterplot.

c) Create an appropriate model.

d) Estimate the stopping distance for a car traveling 55 mph.

e) Estimate the stopping distance for a car traveling 70 mph.

f) How much confidence do you place in these predictions? Why?

Distance from velocity data The accompanying table gives

data for the velocity of a vintage sports car accelerating from 0 to

142 mi/h in 36 sec $(10$ thousandths of an hour).

a. Use rectangles to estimate how far the car traveled during the

36 sec it took to reach 142 mi/h.

b. Roughly how many seconds did it take the car to reach the

halfway point? About how fast was the car going then?

Figure $3-16$ shows the velocity-time graph for an automobile on a test track. Describe how the velocity changes with time. (3.1) CANT COPY THE FIGURE

Constant Acceleration

Cars $A$ and $B$ move in the same direction in adjacent lanes.The position $x$ of car $A$ is given in Fig. $2-30,$ from time $t=0$ to $t=7.0 \mathrm{s}$ The figure's vertical scaling is set by $x_{s}=$

32.0 $\mathrm{m} .$ At $t=0,$ car $B$ is at $x=$ 0 , with a velocity of 12 $\mathrm{m} / \mathrm{s}$ and a negative constant acceleration $a_{B} .$ (a) What must $a_{B}$ be such that the cars are (momentarily side by side (momentarily at the same value of $x )$ at $t=4.0 \mathrm{s}$ ? (b) For that value of $a_{B}$ , how many times are the cars side by side? (c) Sketch the position $x$ of car $B$ versus time $t$ on Fig. $2-30 .$ How many times will the cars be side by side if the magnitude of acceleration $a_{B}$ is $(\mathrm{d})$ more than and $(\mathrm{e})$ less than the answer to part $(\mathrm{a}) ?$

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?

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?

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.

Solve each problem.

The distance that it takes a car to stop is a function of the speed and the drag factor. The drag factor is a measure of the resistance between the tire and the road surface. The formula $S=\sqrt{30 L D}$ is used to determine the minimum speed $S$ [ in miles per hour (mph)] for a car that has left skid marks of length $L$ feet ( $\mathrm{ft}$ ) on a surface with drag factor $D$.

a) Find the minimum speed for a car that has left skid marks of length 50 ft where the drag factor is 0.75

b) Does the drag factor increase or decrease for a road surface when it gets wet?

c) Write $L$ as a function of $S$ for a road surface with drag factor 1 and graph the function.

(GRAPH AND IMAGE CAN'T COPY)

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?.

The following strobe drawings represent the motions of two cars, a and b. During which interval of the motion of car a is the average speed of car a approximately equal to the average speed of car b?

CAN'T COPY THE FIGURE

Two highways intersect as shown in Fig. $4-46 .$ At the instan shown, a police car $P$ is distance $d_{P}=800 \mathrm{m}$ from the intersection

and moving at speed $v_{p}=80 \mathrm{km} / \mathrm{h}$ . Motorist $M$ is distance $d_{M}=$

600 $\mathrm{m}$ from the intersection and moving at speed $v_{M}=60 \mathrm{km} / \mathrm{h}$ .(a) In unit-vector notation, what is the velocity of the motorist

with respect to the police car? (b) For the instant shown in Fig. $4-46$ ,

what is the angle between the velocity found in $(a)$ and the line of sight between the two cars? (c) If the cars maintain their veloci

ties, do the answers to (a) and (b) change as the cars move neare

the intersection?

Distance The next two graphs are from the Road \& Track website. The curves show the velocity at $t$ seconds after the car accelerates from a dead stop. To find the total distance traveled by the car in reaching 130 mph, we must estimate the definite integral

$\int_{0}^{T} v(t) d t$

where $T$ represents the number of seconds it takes for the car to reach 130 $\mathrm{mph} .$

Use the graphs to estimate this distance by adding the areas of rectangles and using the midpoint rule. To adjust your answer to miles per hour, divide by 3600 (the number of seconds in an hour). You then have the number of miles that the car traveled in reaching 130 mph. Finally, multiply by 5280 ft per mile to convert the answer to feet. Source: Road $\&$ Track.

Estimate the distance traveled by the Lamborghini Gallardo L. $\mathrm{P} 560-4$ using the graph below. Use rectangles with widths of 3 seconds, except for the last rectangle, which should have a width of 2 seconds. The circle marks the point where the car has gone

a quarter mile. Does this seem correct?

Two vehicles start out traveling side by side along a straight road. Their position functions, shown in the

following graph, are given by $s=f(t)$ and $s=g(t)$ where $s$ is measured in feet and $t$ is measured in seconds.

a. Which vehicle has traveled farther at $t=2$ seconds?

b. What is the approximate velocity of each vehicle at $t=3$ seconds?

c. Which vehicle is traveling faster at $t=4$ seconds?

d. What is true about the positions of the vehicles at $t-4$ seconds?

A model car starts from rest and travels in a straight line. A smartphone mounted on the car has an app that transmits the magnitude of the car's acceleration (measured by an accelerometer) every second. The results are given in the table:

Each measured value has some experimental error. (a) Plot acceleration versus time and find the equation for the straight line that gives the best fit to the data. (b) Use the equation for $a(t)$ that you found in part (a) to calculate $v(t)$, the speed of the car as a function of time. Sketch the graph of $v$ versus $t$. Is this graph a straight line? (c) Use your result from part (b) to calculate the speed of the car at $t =$ 5.00 s. (d) Calculate the distance the car travels between $t =$ 0 and $t =$ 5.00 s.

Additional Problems

A car moving with constant acceleration covered the distance between two points 60.0 $\mathrm{m}$ apart in 6.00 $\mathrm{s}$ . Its speed as it passed the second point was 15.0 $\mathrm{m} / \mathrm{s}$ . (a) What was the speed at the first point? (b) What was the magnitude of the acceleration? (c) At what prior distance from the first point was the car at rest? (d) Graph $x$ versus $t$ and $v$ versus $t$ for the car, from rest $(t=0)$.

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 $t=4.00 \mathrm{s}.$

The accompanying table shows time-speed data for a sports car accelerating from rest to 130 mph. How far had the car traveled by the time it reached this speed? (Use trapezoids to estimate the area under the velocity curve, but be careful: The time intervals vary in length.)

(TABLE CAN'T COPY)

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 $t=35 \mathrm{s} ?$

While entering a freeway, a car accelerates from rest at a rate of 2.40 $\mathrm{m} / \mathrm{s}^{2}$ for 12.0 s. (a) Draw a sketch of the situation. (b) List the knowns in this problem. (c) How far does the car travel in those 12.0 s? To solve this part, first identify the unknown, and then discuss how you chose the appropriate equation to solve for it. After choosing the equation, show your steps in solving for the unknown, check your units, and discuss whether the answer is reasonable. (d) What is the car's final velocity? Solve for this unknown in the same manner as in part (c), showing all steps explicitly.

While entering a freeway, a car accelerates from rest at a rate of $2.40 \mathrm{m} / \mathrm{s}^{2}$ for $12.0 \mathrm{s}$. (a) Draw a sketch of the situation. (b) List the knowns in this problem. (c) How far does the car travel in those $12.0 \mathrm{s}$ ? To solve this part, first identify the unknown, then indicate how you chose the appropriate equation to solve for it. After choosing the equation, show your steps in solving for the unknown, check your units, and discuss whether the answer is reasonable. (d) What is the car's final velocity? Solve for this unknown in the same manner as in (c), showing all steps explicitly.

While entering a freeway, a car accelerates from rest at a rate of $2.40 \mathrm{m} / \mathrm{s}^{2}$ for $12.0 \mathrm{s}$. (a) Draw a sketch of the situation. (b) List the knowns in this problem. (c) How far does the car travel in those $12.0 \mathrm{s} ?$ To solve this part, first identify the unknown, and then discuss how you chose the appropriate equation to solve for it. After choosing the equation, show your steps in solving for the unknown, check your units, and discuss whether the answer is reasonable. (d) What is the car's final velocity? Solve for this unknown in the same manner as in part (c), showing all steps explicitly.

A vehicle moving along a straight road has distance $f(t)$ from its starting point at time $t .$ Which of the graphs in Figure 2.26 could be $f^{\prime}(t)$ for the following scenarios? (Assume the scales on the vertical axes are all the same.)

(a) A bus on a popular route, with no traffic

(b) A car with no traffic and all green lights

(c) A car in heavy traffic conditions

I. (FIGURE CANNOT COPY)

II. (FIGURE CANNOT COPY)

III. (FIGURE CANNOT COPY)

14.The velocity graph of a car accelerating from rest to a speed

of 120 $\mathrm{km} / \mathrm{h}$ over a period of 30 seconds is shown. Estimate

he distance traveled during this period.

(II) Two cars approach a street corner at right angles to each other (sce Fig. $35 ) .$ Car 1 travels at 35 $\mathrm{km} / \mathrm{h}$ and car 2 at 45 $\mathrm{km} / \mathrm{h}$ . What is the relative velocity of car 1 as scen by car 2$?$ What is the velocity of car 2 relative to car 1$?$

The velocity vs. time graph for an object moving along a straight path is shown in Figure P2.24. (i) Find the average acceleration of the object during the time intervals (a) 0 to 5.0 s, (b) 5.0 s to 15 s, and (c) 0 to 20 s. (ii) Find the instantaneous acceleration at (a) 2.0 s, (b) 10 s, and (c) 18 s.

(a) The graph of a position function of a car is shown, where $ s $ is measured in feet and $ t $ in seconds. Use it to graph the velocity and acceleration of the car. What is the acceleration at $ t = 10 $ seconds?

(b) Use the acceleration curve from part (a) to estimate the jerk at $ t = 10 $ seconds. What are the units for jerk?

The graph of the acceleration $ a(t) $ of a car measured in $ ft/s^2 $ is shown. Use the Midpoint Rule to estimate the increase in the velocity of the car during the six-second time interval.

Speed of a Skidding Car Police use the formula $s=\sqrt{30 f d}$ to estimate the speed $s$ (in mi/h) at which a car is traveling if it skids $d$ feet after the brakes are

applied suddenly. The number $f$ is the coefficient of friction of the road, which is a measure of the "slipperiness" of the road. The table gives some typical estimates for $f .$

(a) If a car skids $65 \mathrm{ft}$ on wet concrete, how fast was it

moving when the brakes were applied?

(b) If a car is traveling at $50 \mathrm{mi} / \mathrm{h}$, how far will it skid on

wet tar?

A diagram representing the motion of two cars is shown in Figure P 1.63. The number near each dot indicates the clock reading in seconds when the car passes that location. (a) Indicate times when the cars have the same

A snowmobile moves according to the velocity-time graph shown in the drawing. What is the snowmobile's average acceleration during each of the segments $A, B,$ and $C$ ?

The position $s$ of a car at time $t$ is given in the following table.

$$\begin{array}{c|c|c|c|c|c|c}\hline t(\mathrm{sec}) & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\\hline s(\mathrm{ft}) & 0 & 0.5 & 1.8 & 3.8 & 6.5 & 9.6 \\\hline\end{array}$$

(a) Find the average velocity over the interval $0 \leq t \leq$ 0.2.

(b) Find the average velocity over the interval $0.2 \leq t \leq$ 0.4.

(c) Use the previous answers to estimate the instantaneous velocity of the car at $t=0.2$.

A velocity-time graph for an object moving along the $x$ axis is shown in Figure $\mathrm{P} 2.13 .$ (a) Plot a graph of the acceleration versus time. Determine the average acceleration of the object $(b)$ in the time interval $t=5.00 \mathrm{s}$ to $t=15.0 \mathrm{s}$ and (c) in the time interval $t=0$ to $t=20.0 \mathrm{s}$

The graph in Figure 2.49$~ v_{x}(\mathrm{m} / \mathrm{s})$ shows the velocity of a motorcycle police officer plotted as a function of time. Find the instantaneous acceleration at times $t=3 \mathrm{s},$ at $t=7 \mathrm{s},$ and at $t=11 \mathrm{s}$

In Problem $2,$ what is the speed of the car at (a) point $A$

(b) point $B,$ and $(c)$ point $C ?$ (d) How high will the car go on the last hill, which is too high for it to cross? (e) If we substitute a second car with twice the mass, what then are the answers to (a)

through (d)?

Distance, Speed, and Time A woman driving a car 14 ft long

is passing a truck 30 ft long. The truck is traveling at 50 milh.

How fast must the woman drive her car so that she can pass

the truck completely in 6 s, from the position shown in figure (a) to the position shown in figure (b)? [Hint: Use feet and

seconds instead of miles and hours.