An object projected vertically upward with initial speed

What is the average velocity over the round trip of an object that is launched straight up from the ground and falls straight back down to the ground?

An object thrown straight up falls back and is caught at the same place it is launched from. Its time of flight is T; its maximum height is H. Neglect air resistance. The correct expression for its average speed for the entire flight is (a) H/T, (b) 0, (c) H/(2T), (d) 2H/T.

Using the information in the previous question, what is its average speed just for the first half of the trip? What is its average velocity for the second half of the trip? (Answer in terms of H and T.)

Give an everyday example of one-dimensional motion where (a) the velocity is westward and the acceleration is eastward, and (b) the velocity is northward and the acceleration is northward.

Stand in the center of a large room. Call the direction to your right positive, and the direction to your left negative. Walk across the room along a straight line, using a constant acceleration to quickly reach a steady speed along a straight line in the negative direction. After reaching this steady speed, keep your velocity negative but make your acceleration positive. (a) Describe how your speed varied as you walked. (b) Sketch a graph of x versus t for your motion. Assume you started at x _ 0. (c) Directly under the graph of Part (b), sketch a graph of vx versus t.

True/false: The displacement always equals the product of the average velocity and the time interval. Explain your choice.

Is the statement for an objects velocity to remain constant, its acceleration must remain zero true or false? Explain your choice.

MULTISTEP Draw careful graphs of the position and velocity and acceleration over the time period for a cart that, in succession, has the following motion. The cart is moving at the constant speed of 5.0 m>s in the _x direction. It 0 _ t _ 30 s SSM passes by the origin at t _ 0.0 s. It continues on at 5.0 for 5.0 s, after which it gains speed at the constant rate of 0.50 each second for 10.0 s. After gaining speed for 10.0 s, the cart loses speed at the constant rate of 0.50 for the next 15.0 s.

True/false: Average velocity always equals one-half the sum of the initial and final velocities. Explain your choice.

Identical twin brothers standing on a horizontal bridge each throw a rock straight down into the water below. They throw rocks at exactly the same time, but one hits the water before the other. How can this be? Explain what they did differently. Ignore any effects due to air resistance.

Dr. Josiah S. Carberry stands at the top of the Sears Tower in Chicago. Wanting to emulate Galileo, and ignoring the safety of the pedestrians below, he drops a bowling ball from the top of the tower. One second later, he drops a second bowling ball. While the balls are in the air, does their separation (a) increase over time, (b) decrease, (c) stay the same? Ignore any effects due to air resistance.

Which of the position-versus-time curves in Figure 2-28 best shows the motion of an object (a) with positive acceleration, (b) with constant positive velocity, (c) that is always at rest, and (d) with negative acceleration? (There may be more than one correct answer for each part of the problem.)

Which of the velocity-versus-time curves in Figure 2-29 best describes the motion of an object (a) with constant positive acceleration, (b) with positive acceleration that is decreasing with time, (c) with positive acceleration that is increasing with time, and (d) with no acceleration? (There may be more than one correct answer for each part of the problem.) SSM

The diagram in Figure 2-30 tracks the location of an object moving in a straight line along the x axis. Assume that the object is at the origin at t _ 0. Of the five times shown, which time (or times) represents when the object is (a) farthest from the origin, (b) at rest for an instant, (c) in the midst of being at rest for a while, and (d) moving away from the origin?

An object moves along a straight line. Its positionversus- time graph is shown in Figure 2-30. At which time or times is its (a) speed at a minimum, (b) acceleration positive, and (c) velocity negative?

For each of the four graphs of x versus t in Figure 2-31 answer the following questions. (a) Is the velocity at time t2 greater than, less than, or equal to the velocity at time t1? (b) Is the speed at time t2 greater than, less than, or equal to the speed at time t1?

True/false: (a) If the acceleration of an object is always zero, then it cannot be moving. (b) If the acceleration of an object is always zero, then its x-versus-t curve must be a straight line. (c) If the acceleration of an object is nonzero at an instant, it may be momentarily at rest at that instant. Explain your reasoning for each answer. If an answer is true, give an example.

Ahard-thrown tennis ball is moving horizontally when it bangs into a vertical concrete wall at perpendicular incidence. The ball rebounds straight back off the wall. Neglect any effects due to gravity for the small time interval described here. Assume that toward the wall is the _x direction. What are the directions of its velocity and acceleration (a) just before hitting the wall, (b) at maximum impact, and (c) just after leaving the wall?

Aball is thrown straight up. Neglect any effects due to air resistance. (a) What is the velocity of the ball at the top of its flight? (b) What is its acceleration at that point? (c) What is different about the velocity and acceleration at the top of the flight if instead the ball impacts a horizontal ceiling very hard and then returns.

An object that is launched straight up from the ground, reaches a maximum height H, and falls straight back down to the ground, hitting it T seconds after launch. Neglect any effects due to air resistance. (a) Express the average speed for the entire trip as a function of H and T. (b) Express the average speed for the same interval of time as a function of the initial launch speed v0.

A small lead ball is thrown directly upward. True or false: (Neglect any effects due to air resistance.) (a) The magnitude of its acceleration decreases on the way up. (b) The direction of its acceleration on its way down is opposite to the direction of its acceleration on its way up. (c) The direction of its velocity on its way down is opposite to the direction of its velocity on its way up.

At t _ 0, object A is dropped from the roof of a building. At the same instant, object B is dropped from a window 10 m below the roof. Air resistance is negligible. During the descent of B to the ground, the distance between the two objects (a) is proportional to t, (b) is proportional to t2, (c) decreases, (d) remains 10 m throughout.

CONTEXT-RICH You are driving a Porsche that accelerates uniformly from 80.5 km/h (50 mi/h) at t _ 0.00 s to 113 km/h (70 mi/h) at t _ 9.00 s. (a) Which graph in Figure 2-32 best describes the velocity of your car? (b) Sketch a position-versus-time graph showing the location of your car during these nine seconds, assuming we let its position x be zero at t _ 0.

Asmall heavy object is dropped from rest and falls a distance D in a time T. After it has fallen for a time 2T, what will be its (a) fall distance from its initial location, (b) its speed, and (c) its acceleration? (Neglect air resistance.)

In a race, at an instant when two horses are running right next to each other and in the same direction (the _x direction), Horse As instantaneous velocity and acceleration are _10 m/s and _2.0 m/s2, respectively, and horse Bs instantaneous velocity and acceleration are _12 m/s and _1.0 m/s2, respectively. Which horse is passing the other at this instant? Explain.

True or false: (a) The equation is always valid for particle motion in one dimension. (b) If the velocity at a given instant is zero, the acceleration at that instant must also be zero. (c) The equation _x _ vav _t holds for all particle motion in one dimension.

If an object is moving in a straight line at constant acceleration, its instantaneous velocity halfway through any time interval is (a) greater than its average velocity, (b) less than its average velocity, (c) equal to its average velocity, (d) half its average velocity, (e) twice its average velocity.

A turtle, seeing his owner put some fresh lettuce on the opposite side of his terrarium, begins to accelerate (at a constant rate) from rest at time t _ 0, heading directly toward the food. Let t1 be the time at which the turtle has covered half the distance to his lunch. Derive an expression for the ratio of t2 to t1, where t2 is the time at which the turtle reaches the lettuce.

The positions of two cars in parallel lanes of a straight stretch of highway are plotted as functions of time in the Figure 2-33. Take positive values of x as being to the right of the origin. Qualitatively answer the following: (a) Are the two cars ever side by side? If so, indicate that time (those times) on the axis. (b) Are they always traveling in the same direction, or are they moving in opposite directions for some of the time? If so, when? (c) Are they ever traveling at the same velocity? If so, when? (d) When are the two cars the farthest apart? (e) Sketch (no numbers) the velocity versus time curve for each car. SSM

A car driving at constant velocity passes the origin at time t _ 0. At that instant, a truck, at rest at the origin, begins to accelerate uniformly from rest. Figure 2-34 shows a qualitative plot of the velocities of truck and car as functions of time. Compare their displacements (from the origin), velocities, and accelerations at the instant that their curves intersect.

Reginald is out for a morning jog, and during the course of his run on a straight track, he has a velocity that depends upon time as shown in Figure 2-35. That is, he begins at rest, and ends at rest, peaking at a maximum velocity vmax at an arbitrary time tmax. A second runner, Josie, runs throughout the time interval t _ 0 to t _ tf at a constant speed vJ, so that each has the same displacement during the time interval. Note: tf is NOT twice tmax, but represents an arbitrary time. What is the relation between vJ and vmax?

Which graph (or graphs), if any, of vx versus t in Figure 2-36 best describes the motion of a particle with (a) positive velocity and increasing speed, (b) positive velocity and zero acceleration, (c) constant nonzero acceleration, and (d) a speed decrease?

Which graph (or graphs), if any, of vx versus t in Figure 2-36 best describes the motion of a particle with (a) negative velocity and increasing speed, (b) negative velocity and zero acceleration, (c) variable acceleration, and (d) increasing speed?

Sketch a v-versus-t curve for each of the following conditions: (a) Acceleration is zero and constant while velocity is not zero. (b) Acceleration is constant but not zero. (c) Velocity and acceleration are both positive. (d) Velocity and acceleration are both negative. (e) Velocity is positive and acceleration is negative. ( f ) Velocity is negative and acceleration is positive. (g) Velocity is momentarily zero but the acceleration is not zero.

Figure 2-37 shows nine graphs of position, velocity, and acceleration for objects in motion along a straight line. Indicate the graphs that meet the following conditions: (a) Velocity is constant, (b) velocity reverses its direction, (c) acceleration is constant, and (d) acceleration is not constant. (e) Which graphs of position, velocity, and acceleration are mutually consistent?

CONTEXT-RICH While engrossed in thought about the scintillating lecture just delivered by your physics professor you mistakenly walk directly into the wall (rather than through the open lecture hall door). Estimate the magnitude of your average acceleration as you rapidly come to a halt.

BIOLOGICAL APPLICATION Occasionally, people can survive falling large distances if the surface they land on is soft enough. During a traverse of the Eigers infamous Nordvand, mountaineer Carlos Ragones rock anchor gave way and he plummeted 500 feet to land in snow. Amazingly, he suffered only a few bruises and a wrenched shoulder. Assuming that his impact left a hole in the snow 4.0 ft deep, estimate his average acceleration as he slowed to a stop (that is, while he was impacting the snow).

When we solve free-fall problems near Earth, its important to remember that air resistance may play a significant role. If its effects are significant, we may get answers that are wrong by orders of magnitude if we ignore it. How can we tell when it is valid to ignore the effects of air resistance? One way is to realize that air resistance increases with increasing speed. Thus, as an object falls and its speed increases, its downward acceleration decreases. Under these circumstances, the objects speed will approach, as a limit, a value called its terminal speed. This terminal speed depends upon such things as the mass and cross-sectional area of the body. Upon reaching its terminal speed, its acceleration is zero. For a typical skydiver falling through the air, a typical terminal speed is about 50 m/s (roughly 120 mph). At half its terminal speed, the skydivers acceleration will be about . Let us take half the terminal speed as a reasonable upper bound beyond which we should not use our constant acceleration free-fall relationships. Assuming the skydiver started from rest, (a) estimate how far, and for how long, the skydiver falls before we can no longer neglect air resistance. (b) Repeat the analysis for a Ping-Pong ball, which has a terminal speed of about 5.0 m/s. (c) What can you conclude by comparing your answers for Parts (a) and (b)?

BIOLOGICAL APPLICATION On June 14, 2005, Asafa Powell of Jamaica set a worlds record for the 100-m dash with a time t _ 9.77 s. Assuming he reached his maximum speed in 3.00 s, and then maintained that speed until the finish, estimate his acceleration during the first 3.00 s.

The photograph in Figure 2-38 is a short-time exposure (1/30 s) of a juggler with two tennis balls in the air. (a) The tennis ball near the top of its trajectory is less blurred than the lower one. 34 g SSM Why is that? (b) Estimate the speed of the ball that he is just releasing from his right hand. (c) Determine how high the ball should have gone above the launch point and compare it to an estimate from the picture. Hint: You have a built-in distance scale if you assume some reasonable value for the height of the juggler.

A rough rule of thumb for determining the distance between you and a lightning strike is to start counting the seconds that elapse (one-Mississippi, two-Mississippi, . . .) until you hear the thunder (sound emitted by the lightning as it rapidly heats the air around it). Assuming the speed of sound is about 750 mi/h, (a) estimate how far away is a lightning strike if you counted about 5 s until you heard the thunder. (b) Estimate the uncertainty in the distance to the strike in Part (a). Be sure to explain your assumptions and reasoning. Hint: The speed of sound depends on the air temperature, and your counting is far from exact!

ENGINEERING APPLICATION (a) An electron in a television tube travels the 16-cm distance from the grid to the screen at an average speed of 4.0 _ 107 m/s. How long does the trip take? (b) An electron in a current-carrying wire travels at an average speed of 4.0 _ 10_5 m/s. How long does it take to travel 16 cm?

Arunner runs 2.5 km, in a straight line, in 9.0 min and then takes 30 min to walk back to the starting point. (a) What is the runners average velocity for the first 9.0 min? (b) What is the average velocity for the time spent walking? (c) What is the average velocity for the whole trip? (d) What is the average speed for the whole trip?

A car travels in a straight line with an average velocity of 80 km/h for 2.5 h and then with an average velocity of 40 km/h for 1.5 h. (a) What is the total displacement for the 4.0-h trip? (b) What is the average velocity for the total trip?

One busy air route across the Atlantic Ocean is about 5500 km. The now-retired Concord, a supersonic jet capable of flying at twice the speed of sound, was used for traveling such routes. (a) Roughly how long did it take for a one-way flight? (Use 343 m/s for the speed of sound.) (b) Compare this time to the time taken by a subsonic jet flying at 0.90 times the speed of sound.

The speed of light, designated by the universally recognized symbol c, has a value, to two significant figures, of 3.0 _ 108 m/s. (a) How long does it take for light to travel from the Sun to Earth, a distance of 1.5 _ 1011 m? (b) How long does it take light to travel from the moon to Earth, a distance of 3.8 _ 108 m?

Proxima Centauri, the closest star to us besides our own Sun, is 4.1 _ 1013 km from Earth. From Zorg, a planet orbiting this star, a Gregor places an order at Tonys Pizza in Hoboken, New Jersey, communicating by light signals. Tonys fastest delivery craft travels at 1.00 _ 10_4c (see Problem 46). (a) How long does it take Gregors order to reach Tonys Pizza? (b) How long does Gregor wait between sending the signal and receiving the pizza? If Tonys has a 1000-years-or-its-free delivery policy, does Gregor have to pay for the pizza?

A car making a 100-km journey travels 40 km/h for the first 50 km. How fast must it go during the second 50 km to average 50 km/h?

CONTEXT-RICH Late in ice hockey games, the team that is losing sometimes pulls their goalkeeper off the ice to add an additional offensive player and increase their chances of scoring. In such cases, the goalie on the opposing team might have an opportunity to score into the unguarded net 55.0 m away. Suppose you are the goaltender for your university team and are in just such a situation. You launch a shot (in hopes of getting your first career goal) on the frictionless ice. You eventually hear a disappointing clang as the puck strikes a goalpost (instead of going in!) exactly 2.50 s later. In this case, how fast did the puck travel? You should assume 343 m/s for the speed of sound.

Cosmonaut Andrei, your co-worker at the International Space Station, tosses a banana at you at a speed of 15 m/s. At exactly the same instant, you fling a scoop of ice cream at Andrei along exactly the same path. The collision between banana and ice cream produces a banana split 7.2 m from your location 1.2 s after the banana and ice cream were launched. (a) How fast did you toss the ice cream? (b) How far were you from Andrei when you tossed the ice cream? (Neglect any effects due to gravity.)

Figure 2-39 shows the position of a particle as a function of time. Find the average velocities for the time intervals a, b, c, and d indicated in the figure.

ENGINEERING APPLICATION It has been found that, on average, galaxies are moving away from Earth at a speed that is proportional to their distance from Earth. This discovery is known as Hubbles law, named for its discoverer, astrophysicist Sir Edwin Hubble. He found that the recessional speed v of a galaxy a distance r from Earth is given by v _ Hr, where H _ 1.58 _ 10_18 s_1 is called the Hubble constant. What are the expected recessional speeds of galaxies (a) 5.00 _ 1022 m from Earth, and (b) 2.00 _ 1025m from Earth? (c) If the galaxies at each of these distances had traveled at their expected recessional speeds, how long ago would they have been at our location?

The cheetah can run as fast as 113 km/h, the falcon can fly as fast as 161 km/h, and the sailfish can swim as fast as 105 km/h. The three of them run a relay with each covering a distance L at maximum speed. What is the average speed of this relay team for the entire relay? Compare this average speed with the numerical average of the three individual speeds. Explain carefully why the average speed of the relay team is not equal to the numerical average of the three individual speeds.

Two cars are traveling along a straight road. Car A maintains a constant speed of 80 km/h and car B maintains a constant speed of 110 km/h. At t _ 0, car B is 45 km behind car A. (a) How much farther will car A travel before car B overtakes it? (b) How much ahead of A will B be 30 s after it overtakes A?

Multistep A car traveling at a constant speed of \(20 \mathrm{~m} / \mathrm{s}\) passes an intersection at time t=0. A second car traveling at a constant speed of \(30 \mathrm{~m} / \mathrm{s}\) in the same direction passes the same intersection \(5.0 \mathrm{~s}\) later. (a) Sketch the position functions \(x_1(t)\) and \(x_2(t)\) for the two cars for the interval \(0 \leq t \leq 20 \mathrm{~s}\). (b) Determine when the second car will overtake the first. (c) How far from the intersection will the two cars be when they pull even? (d) Where is the first car when the second car passes the intersection?

BIOLOGICAL APPLICATION Bats use echolocation to determine their distance from objects they cannot easily see in the dark. The time between the emission of a high-frequency sound pulse (a click) and the detection of its echo is used to determine such distances. A bat, flying at a constant speed of 19.5 m/s in a straight line toward a vertical cave wall, makes a single clicking noise and hears the echo 0.15 s later. Assuming that she continued flying at her original speed, how close was she to the wall when she received the echo? Assume a speed of 343 m/s for the speed of sound.

ENGINEERING APPLICATION A submarine can use sonar (sound traveling through water) to determine its distance from other objects. The time between the emission of a sound pulse (a ping) and the detection of its echo can be used to determine such distances. Alternatively, by measuring the time between successive echo receptions of a regularly timed set of pings, the submarines speed may be determined by comparing the time between echoes to the time between pings. Assume you are the sonar operator in a submarine traveling at a constant velocity underwater. Your boat is in the eastern Mediterranean Sea, where the speed of sound is known to be 1522 m/s. If you send out pings every 2.00 s, and your apparatus receives echoes reflected from an undersea cliff every 1.98 s, how fast is your submarine traveling?

A sports car accelerates in third gear from 48.3 km/h (about 30 mi/h) to 80.5 km/h (about 50 mi/h) in 3.70 s. (a) What is the average acceleration of this car in m/s2? (b) If the car maintained this acceleration, how fast would it be moving one second later?

An object is moving along the x axis. At t _ 5.0 s, the object is at x _ _3.0 m and has a velocity of _5.0 m/s. At t _ 8.0 s, it is at x _ _9.0 m and its velocity is _1.0 m/s. Find its average acceleration during the time interval 5.0 s t 8.0 s.

A particle moves along the x axis with velocity vx _ (8.0 m/s2)t 2 7.0 m/s. (a) Find the average acceleration for two different one-second intervals, one beginning at t _ 3.0 s and the other beginning at t _ 4.0 s. (b) Sketch vx versus t over the interval 0 t 10 s. (c) How do the instantaneous accelerations at the middle of each of the two time intervals specified in Part (a) compare to the average accelerations found in Part (a)? Explain.

MULTISTEP The position of a certain particle depends on time according to the equation , where x is in meters if t is in seconds. (a) Find the displacement and average velocity for the interval . (b) Find the general formula for the displacement for the time interval from t to t _ _t. (c) Use the limiting process to obtain the instantaneous velocity for any time t.

The position of an object as a function of time is given by , where A _ 8.0 m/s2, B _ 6.0 m/s, and C _ 4.0 m. Find the instantaneous velocity and acceleration as functions of time.

The one-dimensional motion of a particle is plotted in Figure 2-40. (a) What is the average acceleration in each of the intervals AB, BC, and CE? (b) How far is the particle from its starting point after 10 s? (c) Sketch the displacement of the particle as a function of time; label the instants A, B, C, D, and E on your graph. (d)At what time is the particle traveling most slowly?

An object projected vertically upward with initial speed v0 attains a maximum height h above its launch point. Another object projected up with initial speed 2v0 from the same height will attain a maximum height of (a) 4h, (b) 3h, (c) 2h, (d) h. (Air resistance is negligible.)

A car traveling along the x axis starts from rest at x _ 50 m and accelerates at a constant rate of 8.0 m/s2. (a) How fast is it going after 10 s? (b) How far has it gone after 10 s? (c) What is its average velocity for the interval ?

An object traveling along the x axis with an initial velocity of _5.0 m/s has a constant acceleration of _2.0 m/s2. When its speed is 15 m/s, how far has it traveled?

An object traveling along the x axis at constant acceleration has a velocity of _10 m/s when it is at x _ 6.0 m and of _15 m/s when it is at x _ 10.0 m. What is its acceleration?

The speed of an object traveling along the x axis increases at the constant rate of _4.0 m/s each second. At t _ 0.0 s, its velocity is _1.0 m/s and its position is _7.0 m. How fast is it moving when its position is _8.0 m, and how much time has elapsed from the start at t _ 0.0 s?

A ball is launched directly upward from ground level with an initial speed of 20 m/s. (Air resistance is negligible.) (a) How long is the ball in the air? (b) What is the greatest height reached by the ball? (c) How many seconds after launch is the ball 15 m above the release point?

In the Blackhawk landslide in California, a mass of rock and mud fell 460 m down a mountain and then traveled 8.00 km across a level plain. It has been theorized that the rock and mud moved on a cushion of water vapor. Assume that the mass dropped with the free-fall acceleration and then slid horizontally, losing speed at a constant rate. (a) How long did the mud take to drop the 460 m? (b) How fast was it traveling when it reached the bottom? (c) How long did the mud take to slide the 8.00 km horizontally? SSM

Aload of bricks is lifted by a crane at a steady velocity of 5.0 m/s when one brick falls off 6.0 m above the ground. (a) Sketch the position of the brick y(t) versus time, from the moment it leaves the pallet until it hits the ground. (b) What is the greatest height the brick reaches above the ground? (c) How long does it take to reach the ground? (d) What is its speed just before it hits the ground?

A bolt comes loose from underneath an elevator that is moving upward at a constant speed of 6.0 m/s. The bolt reaches the bottom of the elevator shaft in 3.0 s. (a) How high above the bottom of the shaft was the elevator when the bolt came loose? (b) What is the speed of the bolt when it hits the bottom of the shaft?

An object is dropped from rest at a height of 120 m. Find the distance it falls during its final second in the air.

An object is released from rest at a height h. During the final second of its fall, it traverses a distance of 38 m. Determine h.

A stone is thrown vertically downward from the top of a 200-m cliff. During the last half second of its flight, the stone travels a distance of 45 m. Find the initial speed of the stone.

An object is released from rest at a height h. It travels 0.4h during the first second of its descent. Determine the average velocity of the object during its entire descent.

A bus accelerates from rest at 1.5 m/s2 for 12 s. It then travels at constant velocity for 25 s, after which it slows to a stop with an acceleration of magnitude 1.5 m/s2. (a) What is the total distance that the bus travels? (b) What is its average velocity?

Al and Bert are jogging side-by-side on a trail in the woods at a speed of 0.75 m/s. Suddenly Al sees the end of the trail 35 m ahead and decides to speed up to reach it. He accelerates at a constant rate of 0.50 m/s2 while Bert continues on at a constant speed. (a) How long does it take Al to reach the end of the trail? (b) Once he reaches the end of the trail, he immediately turns around and heads back along the trail with a constant speed of 0.85 m/s. How long does it take him to meet up with Bert? (c) How far are they from the end of the trail when they meet?

You have designed a rocket to be used to sample the local atmosphere for pollution. It is fired vertically with a constant upward acceleration of 20 m/s2. After 25 s, the engine shuts off and the rocket continues rising (in freefall) for a while. (Air resistance is negligible.) The rocket eventually stops rising and then falls back to the ground. You want to get a sample of air that is 20 km above the ground. (a) Did you reach your height goal? If not, what would you change so that the rocket raches 20 km? (b) Determine the total time the rocket is in the air. (c) Find the speed of the rocket just before it hits the ground.

Aflowerpot falls from a windowsill of an apartment that is on the tenth floor of an apartment building. Aperson in an apartment below, coincidentally in possession of a high-speed highprecision timing system, notices that it takes 0.20 s for the pot to fall past his window, which is 4.0-m from top to bottom. How far above the top of the window is the windowsill from which the pot fell? (Neglect any effects due to air resistance.)

In a classroom demonstration, a glider moves along an inclined track with constant acceleration. It is projected from the low end of the track with an initial velocity. After 8.00 s have elapsed, it is 100 cm from the low end and is moving along the track at a velocity of _15 cm/s. Find the initial velocity and the acceleration.

A rock dropped from a cliff covers one-third of its total distance to the ground in the last second of its fall. Air resistance is negligible. How high is the cliff?

A typical automobile under hard braking loses speed at a rate of about 7.0 m/s2; the typical reaction time to engage the brakes is 0.50 s. A local school board sets the speed limit in a school zone such that all cars should be able to stop in 4.0 m. (a) What maximum speed does this imply for an automobile in this zone? (b) What fraction of the 4.0 m is due to the reaction time?

Two trains face each other on adjacent tracks. They are initially at rest, and their front ends are 40 m apart. The train on the left accelerates rightward at 1.0 m/s2. The train on the right accelerates leftward at 1.3 m/s2. (a) How far does the train on the left travel before the front ends of the trains pass? (b) If the trains are each 150 m in length, how long after the start are they completely past one another, assuming their accelerations are constant?

Two stones are dropped from the edge of a 60-m cliff, the second stone 1.6 s after the first. How far below the top of the cliff is the second stone when the separation between the two stones is 36 m?

A motorcycle officer hidden at an intersection observes a car driven by an oblivious driver who ignores a stop sign and continues through the intersection at constant speed. The police officer takes off in pursuit 2.0 s after the car has passed the stop sign. She accelerates at 4.2 m/s2 until her speed is 110 km/h, and then continues at this speed until she catches the car. At that instant, the car is 1.4 km from the intersection. (a) How long did it take for the officer to catch up to the car? (b) How fast was the car traveling?

At t _ 0, a stone is dropped from the top of a cliff above a lake. Another stone is thrown downward 1.6 s later from the same point with an initial speed of 32 m/s. Both stones hit the water at the same instant. Find the height of the cliff.

A passenger train is traveling at 29 m/s when the engineer sees a freight train 360 m ahead of his train traveling in the same direction on the same track. The freight train is moving at a speed of 6.0 m/s. (a) If the reaction time of the engineer is 0.40 s, what is the minimum (constant) rate at which the passenger train must lose speed if a collision is to be avoided? (b) If the engineers reaction time is 0.80 s and the train loses speed at the minimum rate described in Part (a), at what rate is the passenger train approaching the freight train when the two collide? (c) For both reaction times, how far will the passenger train have traveled in the time between the sighting of the freight train and the collision?

BIOLOGICAL APPLICATION The click beetle can project itself vertically with an acceleration of about 400g (an order of magnitude more than a human could survive!). The beetle jumps by unfolding its 0.60-cm long legs. (a) How high can the click beetle jump? (b) How long is the beetle in the air? (Assume constant acceleration while in contact with the ground and neglect air resistance.)

An automobile accelerates from rest at 2.0 m/s2 for 20 s. The speed is then held constant for 20 s, after which there is an acceleration of _3.0 m/s2 until the automobile stops. What is the total distance traveled?

Consider measuring the free-fall motion of a particle (neglect air resistance). Before the advent of computer-driven data-logging software, these experiments typically employed a wax-coated tape placed vertically next to the path of a dropped electrically conductive object. A spark generator would cause an arc to jump between two vertical wires through the falling object and through the tape, thereby marking the tape at fixed time intervals _t. Show that the change in height during successive time intervals for an object falling from rest follows Galileos Rule of Odd Numbers: _y21 _ 3_y10, _y32 _ 5_y10, . . . , where _y10 is the change in y during the first interval of duration _t, _y21 is the change in y during the second interval of duration _t, etc. SSM SSM

Starting from rest, a particle travels along the x axis with a constant acceleration of _ . At a time 4.0 s following its start, it is at x _ _100 m. At a time 6.0 s later it has a velocity of . Find its position at this later time.

If it were possible for a spacecraft to maintain a constant acceleration indefinitely, trips to the planets of the Solar System could be undertaken in days or weeks, while voyages to the nearer stars would only take a few years. (a) Using data from the tables at the back of the book, find the time it would take for a one-way trip from Earth to Mars (at Mars closest approach to Earth). Assume that the spacecraft starts from rest, travels along a straight line, accelerates halfway at 1 g, flips around, and decelerates at 1 g for the rest of the trip. (b) Repeat the calculation for a 4.1 _ 1013-km trip to Proxima Centauri, our nearest stellar neighbor outside of the Sun. (See Problem 47.)

The Stratosphere Tower in Las Vegas is 1137 ft high. It takes 1 min, 20 s to ascend from the ground floor to the top of the tower using the high-speed elevator. The elevator starts and ends at rest. Assume that it maintains a constant upward acceleration until it reaches its maximum speed, and then maintains a constant acceleration of equal magnitude until it comes to a stop. Find the magnitude of the acceleration of the elevator. Express this acceleration magnitude as a multiple of g (the acceleration due to gravity).

A train pulls away from a station with a constant acceleration of 0.40 m/s2. Apassenger arrives at a point next to the track 6.0 s after the end of the train has passed the very same point. What is the slowest constant speed at which she can run and still catch the train? On a single graph, plot the position versus time curves for both the train and the passenger.

Ball A is dropped from the top of a building of height h at the same instant that ball B is thrown vertically upward from the ground. When the balls collide, they are moving in opposite directions, and the speed of A is twice the speed of B. At what height does the collision occur?

Solve Problem 96 if the collision occurs when the balls are moving in the same direction and the speed of A is 4 times that of B.

Starting at one station, a subway train accelerates from rest at a constant rate of 1.00 m/s2 for half the distance to the next station, then slows down at the same rate for the second half of the journey. The total distance between stations is 900 m. (a) Sketch a graph of the velocity vx as a function of time over the full journey. (b) Sketch a graph of the position as a function of time over the full journey. Place appropriate numerical values on both axes.

A speeder traveling at a constant speed of 125 km/h races past a billboard. A patrol car pursues from rest with constant acceleration of (8.0 km/h)/s until it reaches its maximum speed of 190 km/h, which it maintains until it catches up with the speeder. (a) How long does it take the patrol car to catch the speeder if it starts moving just as the speeder passes? (b) How far does each car travel? (c) Sketch x(t) for each car.

When the patrol car in Problem 99 (traveling at 190 km/h) is 100 m behind the speeder (traveling at 125 km/h), the speeder sees the police car and slams on his brakes, locking the wheels. (a) Assuming that each car can brake at 6.0 m/s2 and that the driver of the police car brakes instantly as she sees the brake lights of the speeder (reaction time _ 0.0 s), show that the cars collide. (b) At what time after the speeder applies his brakes do the two cars collide? (c) Discuss how reaction time would affect this problem.

Leadfoot Lou enters the Rest-to-Rest auto competition, in which each contestants car begins and ends at rest, covering a fixed distance L in as short a time as possible. The intention is to demonstrate driving skills, and to find which car is the best at the total combination of speeding up and slowing down. The course is designed so that maximum speeds of the cars are never reached. (a) If Lous car maintains an acceleration (magnitude) of a during speedup, and maintains a deceleration (magnitude) of 2a during braking, at what fraction of L should Lou move his foot from the gas pedal to the brake? (b) What fraction of the total time for the trip has elapsed at that point? (c) What is the fastest speed Lous car ever reaches? Neglect Lous reaction time, and answer in terms of a and L.

A physics professor, equipped with a rocket backpack, steps out of a helicopter at an altitude of 575 m with zero initial velocity. (Neglect air resistance.) For 8.0 s, she falls freely. At that time, she fires her rockets and slows her rate of descent at 15 m/s2 until her rate of descent reaches 5.0 m/s. At this point, she adjusts her rocket engine controls to maintain that rate of descent until she reaches the ground. (a) On a single graph, sketch her acceleration and velocity as functions of time. (Take upward to be positive.) (b) What is her speed at the end of the first 8.0 s? (c) What is the duration of her slowing-down period? (d) How far does she travel while slowing down? (e) How much time is required for the entire trip from the helicopter to the ground? ( f ) What is her average velocity for the entire trip?

The velocity of a particle is given by vx(t) _ (6.0 m/s2)t _ (3.0 m/s). (a) Sketch v versus t and find the area under the curve for the interval t _ 0 to t _ 5.0 s. (b) Find the position function x(t). Use it to calculate the displacement during the interval t _ 0 to t _ 5.0 s.

Figure 2-41 shows the velocity of a particle versus time. (a) What is the magnitude, in meters, represented by the area of the shaded box? (b) Estimate the displacement of the particle for the two 1-s intervals, one beginning at t _ 1.0 s and the other at t _ 2.0 s. (c) Estimate the average velocity for the interval . (d) The equation of the curve is vx _ (0.50 m/s3)t2. Find the displacement of the particle for the interval by integration and compare this answer with your answer for Part (b). Is the average velocity equal to the mean of the initial and final velocities for this case?

The velocity of a particle is given by vx _ (7.0 m/s3)t2 _ 5.0 m/s. If the particle is at the origin at t0 _ 0, find the position function x(t).

Consider the velocity graph in Figure 2-42. Assuming x _ 0 at t _ 0, write correct algebraic expressions for x(t), vx(t), and ax(t) with appropriate numerical values inserted for all constants.

Figure 2-43 shows the acceleration of a particle versus time. (a) What is the magnitude, in m/s, of the area of the shaded box? (b) The particle starts from rest at t _ 0. Estimate the velocity at t _ 1.0 s, 2.0 s, and 3.0 s by counting the boxes under the curve. (c) Sketch the curve vx versus t from your results for Part (b); then estimate how far the particle travels in the interval t _ 0 to t _ 3.0 s.

Figure 2-44 is a graph of vx versus t for a particle moving along a straight line. The position of the particle at time t _ 0 is x0 _ 5.0 m. (a) Find x for various times t by counting boxes, and sketch x as a function of t. (b) Sketch a graph of the acceleration ax as a function of the time t. (c) Determine the displacement of the particle between t _ 3.0 s and 7.0 s.

CONCEPTUAL Figure 2-45 shows a plot of x versus t for an object moving along a straight line. For this motion, sketch graphs (using the same t axis) of (a) vx as a function of t, and (b) ax as a function of t. (c) Use your sketches to qualitatively compare the time(s) when the object is at its largest distance from the origin to the time(s) when its speed is greatest. Explain why the times are not the same. (d) Use your sketches to qualitatively compare the time(s) when the object is moving fastest to the time(s) when its acceleration is the largest. Explain why the times are not the same. SSM

MULTISTEP The acceleration of a certain rocket is given by ax _ bt, where b is a positive constant. (a) Find the position function x(t) if x _ x0 and vx _ v0x at t _ 0. (b) Find the position and velocity at t _ 5.0 s if x0 _ 0, v0x _ 0 and b _ 3.0 m/s3. (c) Compute the average velocity of the rocket between t _ 4.5 s and 5.5 s at t _ 5.0 s if x0 _ 0, v0x _ 0 and b _ 3.0 m/s3. Compare this average velocity with the instantaneous velocity at t _ 5.0 s.

In the time interval from 0.0 s to 10.0 s, the acceleration of a particle traveling in a straight line is given by ax _ (0.20 m/s3)t. Let to the right be the _x direction. The particle initially has a velocity to the right of 9.5 m/s and is located 5.0 m to the left of the origin. (a) Determine the velocity as a function of time during the interval; (b) determine the position as a function of time during the interval; (c) determine the average velocity between t _ 0.0 s and 10.0 s, and compare it to the average of the instantaneous velocities at the start and ending times. Are these two averages equal? Explain.

Consider the motion of a particle that experiences a variable acceleration given by ax _ a0x _ bt, where a0x and b are constants and x _ x0 and vx _ v0x at t _ 0. (a) Find the instantaneous velocity as a function of time. (b) Find the position as a function of time. (c) Find the average velocity for the time interval with an initial time of zero and arbitrary final time t. (d) Compare the average of the initial and final velocities to your answer to Part (c). Are these two averages equal? Explain.

CONTEXT-RICH You are a student in a science class that is using the following apparatus to determine the value of g. Two photogates are used. (Note: You may be familiar with photogates in everyday living. You see them in the doorways of some stores. They are designed to ring a bell when someone interrupts the beam while walking through the door.) One photogate is located at the edge of a table that is 1.00 m above the floor, and the second photogate is located directly below the first, at a height 0.500 m above the floor. You are told to drop a marble through these gates, releasing it from rest a negligible distance above the upper gate. The upper gate starts a timer as the ball passes through its beam. The second photogate stops the timer when the ball passes through its beam. (a) Prove that the experimental magnitude of free-fall acceleration is given by gexp _ (2_y)/(_t)2, where _y is the vertical distance between the photogates and _t is the fall time. (b) For your setup, what value of _t would you expect to measure, assuming gexp is the standard value (9.81 m/s2)? (c) During the experiment, a slight error is made. Instead of locating the first photogate even with the top of the table, your not-so-careful lab partner locates it 0.50 cm lower than the top of the table. However, she does manage to properly locate the second photogate at a height of 0.50 m above the floor. However, she releases the marble from the same height that it was released from when the photogate was 1.00 m above the floor. What value of gexp will you and your partner determine? What percentage difference does this represent from the standard value of g?

MULTISTEP The position of a body oscillating on a spring is given by x _ A sin vt, where A and v (lower case Greek omega) are constants, A _ 5.0 cm, and v _ 0.175 s_1. (a) Plot x as a function of tfor . (b) Measure the slope of your graph at t _ 0 to find the velocity at this time. (c) Calculate the average velocity for a series of intervals, beginning at t _ 0 and ending at t _ 6.0, 3.0, 2.0, 1.0, 0.50, and 0.25 s. (d) Compute dx/dt to find the velocity at time t _ 0. (e) Compare your results in Parts (c) and (d) and explain why your Part (c) results approach your Part (d) result.

CONCEPTUAL Consider an object that is attached to a horizontally oscillating piston. The object moves with a velocity given by v _ B sin(vt), where B and v (lower case Greek omega) are constants and v is in s_1. (a) Explain why B is equal to the maximum speed vmax. (b) Determine the acceleration of the object as a function of time. Is the acceleration constant? (c) What is the maximum acceleration (magnitude) in terms of v and vmax. (d) At t _ 0, the objects position is known to be x0. Determine the position as a function of time in terms of t, v, x0 and vmax.

Suppose the acceleration of a particle is a function of x, where ax(x) _ (2.0 s_2)x. (a) If the velocity is zero when x _ 1.0 m, what is the speed when x _ 3.0 m? (b) How long does it take the particle to travel from x _ 1.0 m to x _ 3.0 m.

A rock falls through water with a continuously decreasing acceleration. Assume that the rocks acceleration as a function of velocity has the form ay _ g _ bvy where b is a positive constant. (The _y direction is directly downward.) (a) What are the SI units of b? (b) Prove mathematically that if the rock is released from rest at time t _ 0, the acceleration will depend exponentially on time according to ay(t) _ ge2bt. (c) What is the terminal speed for the rock in terms of g and b? (See Problem 38 for an explanation of the phenomenon of terminal speed.)

A small rock sinking through water (see Problem 117) experiences an exponentially decreasing acceleration given by ay(t) _ ge2bt, where b is a positive constant that depends on the shape and size of the rock and the physical properties of the water. Based upon this, find expressions for the velocity and position of the rock as functions of time. Assume that its initial position and velocity are both zero and that the _y direction is directly downward.

SPREADSHEET The acceleration of a skydiver jumping from an airplane is given by where b is a positive constant that depends on the skydivers cross-sectional area and the density of the surrounding atmosphere she is diving through. The _y directions is directly downward. (a) If her initial speed is zero when stepping from a hovering helicopter, show that her speed as a function of time is given by , where vt is the terminal speed (see Problem 38) given by , and is a time-scale parameter. (b) What fraction of the terminal speed is the speed at t _ T. (c) Use a spreadsheet program to graph vy(t) as a function of time, using a terminal speed of 56 m/s (use this value to calculate b and T). Does the resulting curve make sense?

APPROXIMATION Imagine that you are standing at a wishing well, wishing that you knew how deep the surface of the water was. Cleverly, you make your wish. Then you take a penny from your pocket and drop it into the well. Exactly three seconds after you dropped the penny, you hear the sound it made when it struck the water. If the speed of sound is 343 m/s, how deep is the well? Neglect any effects due to air resistance.

CONTEXT-RICH You are driving a car at the 25-mi/h speed limit when you observe the light at the intersection 65 m in front of you turn yellow. You know that at that particular intersection the light remains yellow for exactly 5.0 s before turning red. After you think for 1.0 s, you then accelerate the car at a constant rate. You somehow manage to pass your 4.5-m-long car completely through the 15.0-m-wide intersection just as the light turns red, thus narrowly avoiding a ticket for being in an intersection when the light is red. Immediately after passing through the intersection, you take your foot off the accelerator, relieved. However, down the road you are pulled over for speeding. You assume that you were ticketed for the speed of your car as it exited the intersection. Determine this speed and decide whether you should fight this ticket in court. Explain.

For a spherical celestial object of radius R, the acceleration due to gravity g at a distance x from the center of the object is where g0 is the acceleration due to gravity at the objects surface and . For the moon, take g0 _ 1.63 m/s2 and R _ 3200 km. If a rock is released from rest at a height of 4R above the lunar surface, with what speed does the rock impact the moon? Hint: Its acceleration is a function of position and increases as the object falls. So do not use constant acceleration free-fall equations, but go back to basics.