Mechanics

How many gallons of gasoline are used in the United States in one day? Assume that there are two cars for every three people, that each car is driven an average of $10,000$ miles per year, and that the average car gets $20$ miles per gallon.

The position of the front bumper of a test car under microprocessor control is given by $\mathbf{x}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{17}\mathbf{\hspace{0.33em}}\mathbf{m}\mathbf{+}\mathbf{(}\mathbf{4}\mathbf{.}\mathbf{80}\mathbf{\hspace{0.33em}}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}\mathbf{)}{\mathbf{t}}^{\mathbf{2}}\mathbf{-}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{100}\mathbf{\hspace{0.33em}}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{6}}\mathbf{)}{\mathbf{t}}^{\mathbf{6}}$. (a) Find its position and acceleration at the instants when the car has zero velocity. (b) Draw $\mathbf{x}\mathbf{-}\mathbf{t}\mathbf{,}\mathbf{\hspace{0.33em}}{\mathbf{v}}_{\mathbf{x}}\mathbf{-}\mathbf{t}\mathbf{\hspace{0.33em}}\mathbf{and}\mathbf{\hspace{0.33em}}{\mathbf{a}}_{\mathbf{x}}\mathbf{-}\mathbf{t}$ graphs for the motion of the bumper between $\mathbf{t}\mathbf{=}\mathbf{0}\mathbf{\hspace{0.33em}}\mathbf{and}\mathbf{\hspace{0.33em}}\mathbf{t}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{\hspace{0.33em}}\mathbf{s}$.

An antelope moving with constant acceleration covers the distance between two points $70.0$apart in$6.00s$. Its speed as it passes the second point  is$15.0m/s$ . What are (a) its speed at the first point and (b) its acceleration?

A jet fighter pilot wishes to accelerate from rest at a constant acceleration of to reach Mach 3 (three times the speed of sound) as quickly as possible. Experimental tests reveal that he will black out if this acceleration lasts for more than5.0s . Use $331m/s$ for the speed of sound. (a) Will the period of acceleration last long enough to cause him to black out? (b) What is the greatest speed he can reach with an acceleration of  before he blacks out?

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

A Tennis Serve. In the fastest measured tennis serve, the ball left the racquet at $\mathbf{73}.\mathbf{14} m/s$. A served tennis ball is typically in contact with the racquet for  $\mathbf{30}.\mathbf{0}$and starts from rest. Assume constant acceleration. (a) What was the ball’s acceleration during this serve? (b) How far did the ball travel during the serve?

Automobile Air Bags. The human body can survive an acceleration trauma incident (sudden stop) if the magnitude of the acceleration is less than $\mathbf{250}\mathbf{\hspace{0.33em}}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$. If you are in an automobile accident with an initial speed of $\mathbf{105}\mathbf{ }\mathbf{km}\mathbf{/}\mathbf{h}\mathbf{ }\mathbf{(}\mathbf{65}\mathbf{\hspace{0.33em}}\mathbf{mi}\mathbf{/}\mathbf{h}\mathbf{)}$  and are stopped by an airbag that inflates from the dashboard, over what distance must the airbag stop you for you to survive the crash?

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. 2.13b and 2.14b), show the position, velocity, and acceleration of each of the two cars at \(t = 0,t = 1{\rm{ s}}\) and \(t = 3{\rm{ s}}\).

For the vectors  $\overline{A}$ and $\overrightarrow{B}$ in Fig. E1.24, use a scale drawing to find the magnitude and direction of (a) the vector sum  $\overline{A}$+ $\overrightarrow{B}$ and (b) the vector difference.  $\overline{A}$- $\overrightarrow{B}$ Use your answers to find the magnitude and direction of (c) -$\overline{A}$- $\overrightarrow{B}$ and (d) $\overrightarrow{B}$- $\overline{A}$(See alsoExercise 1.31 for a different approach.)

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

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

Let   be the angle that the vector   makes with the +x-axis, measured counterclockwise from that axis. Find angle   for a vector that has these components: (a) $A_x=2.00m,A_y=-1.00m$ ; (b$A_x=2.00m,A_y=1.00m$ ; (c) $A_x=-2.00m,A_y=1.00m$ ; (d) $A_x=-2.00m,A_y=-1.00m$  

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

Vector $A$ has y-component $Ay = +9.60 m$. A makes an angle of $32.0$ counterclockwise from the +y-axis. (a) What is the x-component of $A$ ? (b) What is the magnitude of $A$ ?

At launch a rocket ship weighs 4.5 million pounds. When it is launched from rest, it takes 8.00 s to reach 161 km/h; at the end of the first 1.00 min, its speed is 1610 km/h. (a) What is the average acceleration (in m/s²) of the rocket (i) during the first 8.00 s and (ii) between 8.00 s and the end of the first 1.00 min? (b) Assuming the acceleration is constant during each time interval (but not necessarily the same in both intervals), what distance does the rocket travel (i) during the first 8.00 s and (ii) during the interval from 8.00 s to 1.00 min?

A disoriented physics professor drives 3.25 km north, then 2.20 km west, and then 1.50 km south. Find the magnitude and direction of the resultant displacement, using the method of components. In a vector-addition diagram (roughly to scale), show that the resultant displacement found from your diagram is in qualitative agreement with the result you obtained by using the method of components.

A small block has constant acceleration as it slides down a frictionless incline. The block is released from rest at the top of the incline, and its speed after it has travelled 6.80 m to the bottom of the incline is 3.80 m/s. What is the speed of the block when it is 3.40 m from the top of the incline?

Find the magnitude and direction of the vector represented by the following pairs of components: (a) $A_x=-8.60 \mathrm{cm},A_y=5.20 \mathrm{cm};$ (b) $A_{\mathrm{x}}=-9.70 \mathrm{m},A_{\mathrm{y}}=-2.45 \mathrm{m}$(c) $A_{\mathrm{x}}=7.75 \mathrm{km},A_{\mathrm{y}}=-2.70 \mathrm{km}$

At the instant the traffic light turns green, a car that has been waiting at an intersection starts ahead with a constant acceleration of 2.80 m/s². At the same instant a truck, travelling with a constant speed of 20.0 m/s, overtakes and passes the car. (a) How far beyond its starting point does the car overtake the truck? (b) How fast is the car travelling when it overtakes the truck? (c) Sketch an x-t graph of the motion of both vehicles. Take x = 0 at the intersection. (d) Sketch a v_x-t graph of the motion of both vehicles.

Vector $\overrightarrow{A} \mathrm{S}$ is $2.80 \mathrm{cm}$ long and is $60°$ above the x-axis in the first quadrant. Vector ${\overrightarrow{B}}_{\text{is }}1.90 \mathrm{cm}$ long and is $60°$ below the $x$-axis in the fourth quadrant (Fig. E1.35). Use components to find the magnitude and direction of (a) $\overrightarrow{A}+\overrightarrow{B}$; (b) $\overrightarrow{A}-\overrightarrow{B}$; (c) $\overrightarrow{B}-\overrightarrow{A}$. In each case, sketch the vector addition or subtraction and show that your numerical answers are in qualitative agreement with your sketch.

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

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

A juggler throws a bowling pin straight up with an initial

speed of 8.20 m/s. How much time elapses until the bowling pin

returns to the juggler’s hand?

You throw a glob of putty straight up toward the ceiling,which is 3.60 m above the point where the putty leaves your hand. The initial speed of the putty as it leaves your hand is 9.50 m/s.

(a) What is the speed of the putty just before it strikes the ceiling?

(b) How much time from when it leaves your hand does it take the putty to reach the ceiling?

A lunar lander is makingits descent to Moon Base I (Fig. E2.40). The lander descendsslowly under the retro-thrust of its descent engine. The engine iscut off when the lander is 5.0 m above the surface and has a downwardspeed of 0.8m/s . With the engine off, the lander is in freefall. What is the speed of the lander just before it touches the surface?The acceleration due to gravity on the moon is $1.6m/s^2$ .

A lunar lander is making its descent to Moon Base I (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$\mathbf{0}\mathbf{.}\mathbf{8}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$. 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$\mathbf{1}\mathbf{.}\mathbf{6}\mathbf{ }\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$.

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

The rocket-driven sled Sonic Wind No. 2, used for investigating the physiological effects of large accelerations, runs on a straight, level track 1070 m (3500 ft) long. Starting from rest, it can reach a speed of 224 m/s (500 mi/h) in 0.900 s. 

(a) Compute the acceleration in $\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$, assuming that it is constant. 

(b) What is the ratio of this acceleration to that of a freely falling body (g)? (c) What distance is covered in 0.900 s? 

(d) A magazine article states that at the end of a certain run, the speed of the sled decreased from 283 m/s (632 mi/h) to zero in 1.40 s and that during this time the magnitude of the acceleration was greater than 40g. Are these figures consistent?

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

You throw a small rock straight up from the edge of a highway bridge that crosses a river. The rock passes you on its way down, 6.00 s after it was thrown. What is the speed of the rock just before it reaches the water 28.0 m below the point where the rock left your hand? Ignore air resistance.

A small object moves along the x-axis with acceleration a_x(t) = -(0.0320 m/s³)(15.0 s – t). At t = 0 the object is at x = -14.0 m and has velocity v_0x = 8.00 m/s. What is the x-coordinate of the object when t = 10.0 s?

A rocket starts from rest and moves upward from the surface of the earth. For the first 10.0 s of its motion, the vertical acceleration of the rocket is given by a_y = (2.80 m/s³) t, where the +y-direction is upward. (a) What is the height of the rocket above the surface of the earth at t = 10.0 s? (b) What is the speed of the rocket when it is 325 m above the surface of the earth?

The density of air under standard laboratory conditions is$\mathbf{1}\mathbf{.}\mathbf{29} \mathbf{kg}\mathbf{/}{\mathbf{m}}^{\mathbf{3}}$ , and about 20% of that air consists of oxygen. Typically, people breathe about 1/2 L of air per breath. (a) How many grams of oxygen does a person breathe in a day? (b) If this air is stored uncompressed in a cubical tank, how long is each side of the tank?

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

As you eat your way through a bag of chocolate chip cookies, you observe that each cookie is a circular disk with a diameter of $\mathbf{8}\mathbf{.}\mathbf{50}\mathbf{\pm }\mathbf{0}\mathbf{.}\mathbf{02}$cm cm and a thickness of$\mathbf{0}\mathbf{.}\mathbf{050}\mathbf{\pm }\mathbf{0}\mathbf{.}\mathbf{005}\mathbf{cm}$ (a) Find the average volume of a cookie and the uncertainty in the volume. (b) Find the ratio of the diameter to the thickness and the uncertainty in this ratio.

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 $\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{b}\mathbf{-}\mathbf{ct}\mathbf{+}{\mathbf{dt}}^{\mathbf{2}}$, where $\mathbf{d}\mathbf{=}\mathbf{800}\mathbf{ }\mathbf{m}$ is the initial height of the lander above the surface, c=6.0 m/s , and d = 1.05 m/s². (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?

Earthquakes produce several types of shock waves. The most well known are the P-waves (P for primary or pressure) and the S-waves (S for secondary or shear). In the earth’s crust, P-waves travel at about 6.5 km/s and S-waves move at about 3.5 km/s. The time delay between the arrival of these two waves at a seismic recording station tells geologists how far away an earthquake occurred. If the time delay is , how far from the seismic station did the earthquake occur?

A brick is dropped from the roof of a tall building. After it has been falling for a few seconds, it falls 40.0 m in a 1.00-s time interval. What distance will it fall during the next 1.00 s? Ignore air resistance.

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

Three horizontal ropes pull on a large stone stuck in the ground, producing the vector forces $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{A}}$ , $\stackrel{\rightharpoonup }{\mathbf{B}}$, and $\stackrel{\rightharpoonup }{\mathbf{C}}$ shown in Fig. P1.60. Find the magnitude and direction of a fourth force on the stone that will make the vector sum of the four forces zero.

A subway train starts from rest at a station and accelerates at a rate of $\mathbf{160}\frac{\mathbf{m}}{{\mathbf{s}}^{\mathbf{2}}}$ for 14s. It runs at constant speed for 70.0 s and slows down at a rate of $\mathbf{3}\mathbf{.}\mathbf{50}\frac{\mathbf{m}}{{\mathbf{s}}^{\mathbf{2}}}$ until it stops at the next station. Find the total distance covered.

As noted in Exercise 1.26, a spelunker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction $45°$ east of south, and then 280 m at $30°$ east of north. After a fourth displacement, she finds herself back where she started. Use the method of components to determine the magnitude and direction of the fourth displacement. Draw the vector-addition diagram and show that it is in qualitative agreement with your numerical solution.

A gazelle is running in a straight line (the x-axis). The graph in Fig. P2.61 shows this animal’s velocity as a function of time. During the first 12.0 s, find (a) the total distance moved and (b) the displacement of the gazelle. (c) Sketch an ${\mathit{a}}_{\mathbf{x}}\mathbf{-}\mathit{t}$ graph showing this gazelle’s acceleration as a function of time for the first 12.0 s.

A plane leaves the airport in Galisteo and flies 170 km at $68°$ east of north; then it changes direction to fly 230 km at 36$°$ south of east, after which it makes an immediate emergency landing in a pasture. When the airport sends out a rescue crew, in which direction and how far should this crew fly to go directly to this plane?

The engineer of a passenger train traveling at 25.0 m/s sights a freight train whose caboose is 200 m ahead on the same track (Fig. P2.62). The freight train is traveling at 15.0 m/s in the same direction as the passenger train. The engineer of the passenger train immediately applies the brakes, causing a constant acceleration of $\mathbf{0}\mathbf{.}\mathbf{100}\mathbf{ }\mathit{m}\mathbf{/}{\mathit{s}}^{\mathbf{2}}$ in a direction opposite to the train’s velocity, while the freight train continues with constant speed. Take x = 0 at the location of the front of the passenger train when the engineer applies the brakes. (a) Will the cows nearby witness a collision? (b) If so, where will it take place? (c) On a single graph, sketch the positions of the front of the passenger train and the back of the freight train.

Figure P2.62

 A patient with a dislocated shoulder is put into a traction apparatus as shown in Fig. P1.63.The pulls  $\overrightarrow{A}$and $\overrightarrow{B}$ have equal magnitudes and must combine to produce an outward traction force of 12.8 N on the patient’s arm.

How large should these pulls be?

A ball starts from rest and rolls down a hill with uniform acceleration, traveling 200 m during the second 5.0 s of its motion. How far did it roll during the first 5.0 s of motion?

A sailor in a small sailboat encounters shifting winds. She sails 2.00 km east, next 3.50 km southeast, and then an additional distance in an unknown direction. Her final position is 5.80 km directly east of the starting point (Fig. P1.64). Find the magnitude and direction of the third leg of the journey. Draw the vector-addition diagram and show that it is in qualitative agreement with your numerical solution.

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

A car and a truck start from rest at the same instant, with the car initially at some distance behind the truck. The truck has a constant acceleration of $20m/s^2$, and the car has an acceleration of$\mathbf{3}\mathbf{.}\mathbf{40}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$ . The car overtakes the truck after the truck has moved$60.0m$ . (a) How much time does it take the car to overtake the truck? (b) How far was the car behind the truck initially? (c) What is the speed of each when they are abreast? (d) On a single graph, sketch the position of each vehicle as a function of time. Take  $x=0$at the initial location of the truck.