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

it goes. So this is Ah, question 20 in chapter two and section one. And it gives us the projectile motion off this object and asks us to estimate the slopes of these four seeking lines over here. Pick you one through, pick you for. And so to find the slope of a Sikh in line in between two points, we first have to calculate the rise and the run. And this is just the vertical distance and horizontal distance from one point to the other. And so the resulting slope is just equal to the rise over the run. So, for a I'm gonna start with pick you one. And, um, we're going from Q one over here to pee just like that. So we're gonna have to find the rise and then the run. We just clear that away. So the rise over here would be 80 minus 20. It's the height off p minus the height of Q one. And then, um, the, um distance along the x axis off p minus that of Q on. So that would be 10 minus five, which is 60/5, which gives those 12 meters per second. And I'm gonna calculate, um, and include the units. And we'll see why in would be we'll see why, Leader. So I'm gonna continue with pick. You two be maintained. Position so and the system 80. But que Tu is now at 38. And we have 10 minus, uh seven, which gives us 42/3 42 meters for every three seconds, which is 14 meters per second p Q three, we have 80 minus 57 over 10 minus 8.5, which gives us 23 meters divided by 1.5 seconds, which is 15.33 meters per second p Q. For is again 80 minus looks to be 72. Divide by turn, minus lam from five, which gives us eight meters for every 0.5 seconds, which is 16 meters per second. Simple. So these are all the slopes off the secret lines. And so for B, they want us to It asks about how fast was it going when it hit the surface? Now here they want us to estimate the speed of the object right before hit the surface. And, um, the most accurate representation of that we have is this speed of the object right before hit the surface. So now pee represents the surface of the moon and the speed that we have The closest speed we have to when the object hit the surface is P Q for. And as you can see, we already calculated that over here. So the answer is simply, just pick you for just 16 years per second. And now you see why? Um, I included the units for every calculation. Because all of these slopes every here represent speeds in this scenario, which fits for our answer for part B, and that's it.

## Recommended Questions

The accompanying figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance $80 \mathrm{m}$ to the surface of the moon. (FIGURE CAN'T COPY)

a. Estimate the slopes of the secant lines $P Q_{1}, P Q_{2}, P Q_{3},$ and $P Q_{4},$ arranging them in a table like the one in Figure 2.6

b. About how fast was the object going when it hit the surface?

The accompanying figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance 80 $\mathrm{m}$ to the surface of the moon.

\begin{equation}

\begin{array}{l}{\text { a. Estimate the slopes of the secant lines } P Q_{1}, P Q_{2}, P Q_{3}, \text { and }} \\ {P Q_{4}, \text { arranging them in a table like the one in Figure } 2.6 \text { . }} \\ {\text { b. About how fast was the object going when it hit the surface? }}\end{array}

\end{equation}

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

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

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

Lunar Data The accompanying figure shows a distance-time graph for a wrench that fell from the top platform of a communication mast on the moon to the station roof 80 m below.In

Exercises $9-12,$ at the indicated point find

(a) the slope of the curve,

(b) an equation of the tangent, and

(c) an equation of the normal.

(d) Then draw a graph of the curve, tangent line, and normal line in the same square viewing window

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

Near the surface of the moon, the distance that an object falls is a function of time. It is given by

$d(t)=2.6667 t^{t},$ where $t$ is in seconds and $d(t)$ is in feet. If an object is dropped from a certain

height, find the average velocity of the object from $t=1$ to $t=2$ .

Lunar Gravity On the moon, the acceleration of a free-falling object is $a(t)=-1.6$ meters per second per

second. A stone is dropped from a cliff on the moon and hits the surface of the moon 20 seconds later. How far did it fall? What was its velocity at impact?

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

Near the surface of the moon, the distance that an object falls is a function of time. It is given by $d(t)=2.6667 t^{2}$ , where $t$ is in seconds and $d(t)$ is in feet. If an object is dropped from a certain height, find the average velocity of the object from $t=1$ to $t=2 .$

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 $t=20 \mathrm{sec}$

Landing on the Moon, a spacecraft fires its rockets and comes to a complete stop just $12 \mathrm{m}$ above the lunar surface. It then drops freely to the surface. How long does it take to fall, and what's its impact speed? (Hint: Consult Appendix E.)

The Lunar Module could make a safe landing if its vertical velocity at impact is 3.0 $\mathrm{m} / \mathrm{s}$ or less. Suppose that you want to determine the greatest height $h$ at which the pilot could shut off the engine if the velocity of the lander relative to the surface is $(a)$ zero; $(b) 2.0 \mathrm{m} / \mathrm{s}$ downward; $(c) 2.0 \mathrm{m} / \mathrm{s}$ upward. Use conservation of energy to determine $h$ in

each case. The acceleration due to gravity at the surface of the Moon is 1.62 $\mathrm{m} / \mathrm{s}^{2}$ .

A $1.14 \times 10^{4}$ -kg lunar landing craft is about to touch down on the surface of the moon, where the acceleration due to gravity is 1.60 $\mathrm{m} / \mathrm{s}^{2}$ . At an altitude of 165 $\mathrm{m}$ the craft's downward velocity is 18.0 $\mathrm{m} / \mathrm{s}$ . To slow down the craft, a retrorocket is firing to provide an upward thrust. Assuming the descent is vertical, find the magnitude of the thrust needed to reduce the velocity to zero at the instant when the craft touches the lunar surface.

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.

Orbit of a Satellite A satellite is in orbit around the moon.

A coordinate plane containing the orbit is set up with the

center of the moon at the origin, as shown in the graph,

with distances measured in megameters (Mm). The equation

of the satellite's orbit is

$$

\frac{(x-3)^{2}}{25}+\frac{y^{2}}{16}=1

$$

(a) From the graph, determine the closest and the farthest

that the satellite gets to the center of the moon.

(b) There are two points in the orbit with $y$ -coordinates

2. Find the $x$ -coordinates of these points, and determine

their distances to the center of the moon.

Whenever two Apollo astronauts were on the surface of the Moon, a third astronaut orbited the Moon. Assume the orbit to be circular and 100 $\mathrm{km}$ above the surface of the Moon, where the acceleration due to gravity is 1.52 $\mathrm{m} / \mathrm{s}^{2}$ . The radius of the Moon is $1.70 \times 10^{6} \mathrm{m} .$ Determine (a) the astronaut's orbital speed and (b) the period of the orbit.

(II) For an object falling freely from rest, show that the distance traveled during each successive second increases in the ratio of successive odd integers $(1,3,5,$ etc. $)$ . (This was first shown by Galileo.) See Figs. 26 and $29 .$ FIGURE 26 Multiflash photograph of a falling apple, at equal time intervals. The apple falls farther during each successive interval, which means it is accelerating.

How much time did it take for the lander to drop the final 4.30 $\mathrm{ft}$

to the Moon's surface?

\begin{equation}\begin{array}{ll}{\text { A. } 1.18 \mathrm{s}} & {\text { B. } 1.37 \mathrm{s}} \\ {\text { C. } 1.78 \mathrm{s}} & {\text { D. } 2.36 \mathrm{s}}\end{array}\end{equation}

A feather dropped on the moon On the moon, a feather will fall to the ground at the same rate as a heavy stone. Suppose a feather is dropped from a height of $40 \mathrm{m}$ above the surface of the moon. Its height (in meters) above the ground after $t$ seconds is $s=40-0.8 t^{2} .$ Determine the velocity and acceleration of the feather the moment it strikes the surface of the moon.

Free-Fall Acceleration

A ball is shot vertically upward from the surface of another planet. A plot of $y$ versus $t$ for the ball is shown in Fig. $2-36,$ where $y$ is the height of the ball above its starting point and $t=0$ at the instant the ball is shot. The figure's vertical scaling is set by $y_{s}=30.0 \mathrm{m} .$ What are the magnitudes of (a) the free-fall acceleration on the planet and (b) the initial velocity of the ball?

The free-fall acceleration on the surface of the Moon is about one-sixth that on the surface of the Earth. The radius of the Moon is about 0.250$R_{E}\left(R_{E}=\text { Earth's radius }=\right.$ $6.37 \times 10^{6} \mathrm{m}$ ). Find the ratio of their average densities, $\rho_{\text { Moon }} / \rho_{\text { Earth }}$

The table shows the (downward) velocity of a falling object. Estimate the distance fallen. $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|}

\hline \text { time (s) } & 0 & 0.5 & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 & 3.5 & 4.0 \\

\hline \text { velocity (m/s) } & 10 & 14.9 & 19.8 & 24.7 & 29.6 & 34.5 & 39.4 & 44.3 & 49.2 \\

\hline

\end{array}$$

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

An astronaut on the surface of the Moon fires a cannon to launch an experiment package, which leaves the barrel moving horizontally. Assume the free-fall acceleration on the Moon is one-sixth of the package be so that it travels completely around the Moon and returns to its original location? (b) What time interval does this trip around the Moon require?

A spacecraft is in a circular orbit about the moon, $1.22 \times 10^{5} \mathrm{m}$ above its surface. The speed of the spacecraft is $1620 \mathrm{m} / \mathrm{s},$ and the radius of the moon is $1.74 \times 10^{6} \mathrm{m} .$ If the moon were a smooth, reflective sphere, (a) how far below the moon’s surface would the image of the spacecraft appear, and (b) what would be the apparent speed of the spacecraft’s image? (Hint: Both the spacecraft and its image have the same angular speed about the center of the moon.)

Group Aetivity Lumar Module A lunar excursion module is in a circular orbit 250 $\mathrm{km}$ above the surface of the Moon. Assume that the Moon's radius is 1740 $\mathrm{km}$ and that

$k=0.012$ . Find the following.

(a) the velocity of the lunar module

(b) the length of time required for the lunar module to circle the Moon once

An astronaut standing on the moon’s surface throws a rock upward with an initial velocity of 50 feet per second. The height of the rock can be modeled by $m=-2.7 t^{2}+50 t+6,$ where m is the height of the rock (in feet) and t is the time (in seconds). If the astronaut throws the same rock upward with the same initial velocity on Earth, the height of the rock is modeled by $e=-16 t^{2}+50 t+6.$ Would the rock hit the ground in less time on the moon or on Earth? Explain your answer.

Free fall on the moon On our moon, the acceleration of gravity is 1.6 $\mathrm{m} / \mathrm{sec}^{2} .$ If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 30 sec later?

The "Vomit Comet." In microgravity astronaut training and equipment testing, NASA flies a KCl35A aircraft along a parabolic flight path. As shown in Figure P4.47, the aircraft climbs from 24000 $\mathrm{ft}$ to 31000 $\mathrm{ft}$ , where it enters a paraded cabin are in free fall; astronauts and equipment float freely as if there were no gravity. What are the aircraft's (a) speed and (b) altitude at the top of the maneuver? (c) What is the time interval spent in microgravity?

The Moon's speed in its orbit is approximately $1 \mathrm{km} / \mathrm{s}$ and the Earth-Moon distance is $380,000 \mathrm{km}$. Show that these numbers yield an acceleration for the Moon that is very close to that given in the text.

Free Fall with Air Resistance An object is dropped straight down from a helicopter. The object falls faster and faster but its acceleration (rate of change of its velocity) decreases over time because of air resistance. The acceleration is measured in$\mathrm{ft} / \mathrm{sec}^{2}$ and recorded every second after the drop for $5 \mathrm{sec},$ as shown in the table below.

$$\begin{array}{c|ccccc}{t} & {0} & {1} & {2} & {3} & {4} & {5} \\ \hline a & {32.00} & {19.41} & {11.77} & {7.14} & {4.33} & {2.63}\end{array}$$

(a) Use $L R A M_{5}$ to find an upper estimate for the speed when $t$=5.

(b) Use RRAM $_{5}$ to find a lower estimate for the speed when $t=5 .$

(c) Use upper estimates for the speed during the first second, second second, and third second to find an upper estimate for the distance fallen when $t=3 .$

What was the impact speed of the lander when it touched

down? Give your answer in feet per second (ft/s), the same units

used by the astronauts.

\begin{equation}\begin{array}{ll}{\text { A. } 2.41 \mathrm{ft} / \mathrm{s}} & {\text { B. } 6.78 \mathrm{ft} / \mathrm{s}} \\ {\text { C. } 9.95 \mathrm{ft} / \mathrm{s}} & {\text { D. } 10.6 \mathrm{ft} / \mathrm{s}}\end{array}\end{equation}

Astronauts on a distant planet throw a rock straight upward and record its motion with a video camera. After digitizing their video, they are able to produce the graph of height, $y$ ,versus time, $t,$ shown in FIGURE $2-46$ . (a) What is the acceleration of gravity on this planet? (b) What was the initial speed of the rock?

Suppose, instead of shutting off the engine, the astronauts

had increased its thrust, giving the lander a small, but constant,

upward acceleration. Which speed-versus-time plot in Figure

$2-48$ (b) would describe this situation?

\begin{equation}A \quad B \quad C \quad D\end{equation}

Suppose an astronaut drops a feather from $1.2 \mathrm{m}$ above the surface of the Moon. If the acceleration due to gravity on the Moon is $1.62 \mathrm{m} / \mathrm{s}^{2}$ downward, how long does it take the feather to hit the Moon's surface?

Lunar projectile motion A rock thrown vertically upward

from the surface of the moon at a velocity of 24 $\mathrm{m} / \mathrm{sec}$ (about

86 $\mathrm{km} / \mathrm{h} )$ reaches a height of $s=24 t-0.8 t^{2} \mathrm{m}$ in $t \mathrm{sec}$ .

a. Find the rock's velocity and acceleration at time $t$ . The

acceleration in this case is the acceleration of gravity on the

moon.)

b. How long does it take the rock to reach its highest point?

c. How high does the rock go?

d. How long does it take the rock to reach half its maximum

height?

e. How long is the rock aloft?

An unmanned spacecraft is in a circular orbit around the moon, observing the lunar surface from an altitude of 50.0 km (see Appendix F). To the dismay of scientists on earth, an electrical fault causes an on-board thruster to fire, decreasing the speed of the spacecraft by 20.0 m/s. If nothing is done to correct its orbit, with what speed (in km/h) will the spacecraft crash into the lunar surface?

The End of the Lunar Module On Apollo Moon missions, the lunar module would blast off from the Moon's surface and dock with the command module in lunar orbit. After docking, the lunar module

would be jettisoned and allowed to crash back onto the lunar surface. Seismometers placed on the Moon's surface by the astronauts would then pick up the resulting seismic waves. Find the impact

speed of the lunar module, given that is jettisoned from an orbit 110 km above the lunar surface moving with a speed of 1630 $\mathrm{m} / \mathrm{s}$ .

Velocity and Acceleration When an object is dropped straight down, the distance (in feet) that it travels in $t$ seconds is given by

$$s(t)=-16 t^{2} \text { . }$$

Find the velocity at each of the following times.

a. After 3 seconds $\quad$ b. After 5 seconds

c. After 8 seconds

d. Find the acceleration. (The answer here is a constant—the acceleration due to the influence of gravity alone near the surface of Earth.)

As viewed from the surface of the Earth $(A)$, the angle subtended by the full Moon $(D A E)$ is $0.5182^{\circ} .$ Given that the distance from the Earth's surface to the Moon's surface $(A B)$ is approximately 383500 kilometres, find the radius, $r,$ of the Moon to three significant figures.

(FIGURE CAN'T COPY)

An object is released from rest at an altitude $h$ above the surface of the Earth. (a) Show that its speed at a distance $r$ from the Earth's center, where $R_{\mathrm{E}} \leq r \leq R_{E}+h,$ is

$$v=\sqrt{2 G M_{E}\left(\frac{1}{r}-\frac{1}{R_{E}+h}\right)}$$

(b) Assume the release altitude is 500 km. Perform the integral

$$\Delta t=\int_{i}^{f} d t=-\int_{i}^{f} \frac{d r}{v}$$

to find the time of fall as the object moves from the release point to the Earth’s surface. The negative sign appears because the object is moving opposite to the radial direction, so its speed is $v=-d r / d t .$ Perform the integral numerically.

(II) An object starts from rest and falls under the influence of gravity. Draw graphs of ($a$) its speed and ($b$) the distance it has fallen, as a function of time from $t =$ 0 to $t =$ 5.00 s. Ignore air resistance.

Orbit of a Satellite A satellite is in orbit around the moon.

A coordinate plane containing the orbit is set up with the

center of the moon at the origin, as shown in the following

graph, with distances measured in megameters (Mm).

The equation of the satellite's orbit is

$$

\frac{(x-3)^{2}}{25}+\frac{y^{2}}{16}=1

$$

$$

\begin{array}{l}{\text { (a) From the graph, determine the closest to and the far- }} \\ {\text { thest from the center of the moon that the satellite gets. }} \\ {\text { (b) There are two points in the orbit with } y \text { -coordinates } 2 \text { . }} \\ {\text { Find the } x \text { -coordinates of these points, and determine }} \\ {\text { their distances to the center of the moon. }}\end{array}

$$

An unmanned spacecraft is in a circular orbit around the moon, observing the lunar surface from an altitude of 50.0 $\mathrm{km}$ (see Appendix $\mathrm{F}$ ). To the dismay of scientists on earth, an electrical fault causes an on-board thruster to fire, decreasing the speed of the spacecraft by 20.0 $\mathrm{m} / \mathrm{s}$ . If nothing is done to correct its orbit, with what speed (in $\mathrm{km} / \mathrm{h}$ ) will the spacecraft crash into the lunar surface?

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}

A person riding in a subway train drops a ball from rest straight

downward, relative to the interior of the train. The train is moving horizontally with a constant speed of 6.7 $\mathrm{m} / \mathrm{s} .$ A second person standing at rest on the subway platform observes the ball drop. From the point of view of the person on the platform, the ball is released at the position $x=0$ and $y=1.2 \mathrm{m} .$ Make a plot

of the position of the ball for the times $t=0,0.1 \mathrm{s}, 0.2 \mathrm{s}, 0.3$

$\mathrm{s},$ and 0.4 $\mathrm{s} .$ (Your plot is parabolic in shape, as we shall see in

Chapter 4.)

Lunar Lander The Lunar Lander of mass $2.0 \times 10^{4} \mathrm{kg}$

made the last 150 $\mathrm{m}$ of its trip to the Moon's surface in 120 $\mathrm{s}$

descending at approximately constant speed. The Handbook

of Lunar Pilots indicates that the gravitational constant on the

Moon is 1.633 $\mathrm{N} / \mathrm{kg}$ . Using these quantities, what can you

learn about the Lunar Lander's motion?

Falling Distance. (Use $4.9 t^{2}+v_{0} t=s$.)

a) A bolt falls off an airplane at an altitude of $500 \mathrm{m}$. Approximately how long does it take the bolt to reach the ground?

b) A ball is thrown downward at a speed of $30 \mathrm{m} / \mathrm{sec}$ from an altitude of $500 \mathrm{m}$. Approximately how long does it take the ball to reach the ground?

c) Approximately how far will an object fall in 5 sec, when thrown downward at an initial velocity of

$30 \mathrm{m} / \mathrm{sec}$ from a plane?

(II) Estimate by what factor a person can jump farther on the Moon as compared to the Earth if the takeoff speed and angle are the same. The acceleration due to gravity on the Moon is one-sixth what it is on Earth.

At time $t=0,$ a ball is struck at ground level and sent over

level ground. The momentum $p$ versus $t$ during the flight is given by Fig.

$9-46$ (with $p_{0}=6.0 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$ and $p_{1}=4.0 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s} ) .$ At what initial

angle is the ball launched? (Hint:

Find a solution that does not

require you to read the time of the

low point of the plot.)

Distance to the Moon To find the distance to the sun

as in Exercise $67,$ we needed to know the distance to the

moon. Here is a way to estimate that distance: When the

moon is seen at its zenith at a point $A$ on the earth, it is observed to be at the horizon from point $B$ (see the following

figure). Points $A$ and $B$ are 6155 mi apart, and the radius of

the earth is 3960 $\mathrm{mi}$ .

(a) Find the angle $\theta$ in degrees.

(b) Estimate the distance from point $A$ to the moon.

The distance in feet that an object falls in the absence of air resistance is given by $s(t)=16 t^{2},$ where $t$ is time in seconds.

(A) Find $s(0), s(1), s(2),$ and $s(3)$

(B) Find and simplify $\frac{s(2+h)-s(2)}{h}$

(C) Evaluate the expression in part B for $h=\pm 1,\pm 0.1,\pm 0.01,\pm 0.001$

(D) What happens in part $C$ as $h$ gets closer and closer to $0 ?$ What do you think this tells us about the motion of the object? [Hint: Think about what each of the numerator and denominator represents.]