Galileo studied accelerated motion by allowing objects

A person travels by car from one city to another with dif- ferent constant speeds between pairs of cities. She drives for 30.0 min at 80.0 km/h, 12.0 min at 100 km/h, and 45.0 min at 40.0 km/h and spends 15.0 min eating lunch and buying gas. (a) Determine the average speed for the trip. (b) Determine the distance between the initial and final cities along the route.

(a) Sand dunes on a desert island move as sand is swept up the windward side to settle in the leeward side. Such walking dunes have been known to travel 20 feet in a year and can travel as much as 100 feet per year in partic- ularly windy times. Calculate the average speed in each case in m/s. (b) Fingernails grow at the rate of drifting continents, about 10 mm/yr. Approximately how long did it take for North America to separate from Europe, a dis- tance of about 3 000 mi?

Two boats start together and race across a 60-km-wide lake and back. Boat A goes across at 60 km/h and returns at 60 km/h. Boat B goes across at 30 km/h, and its crew, realizing how far behind it is getting, returns at 90 km/h. Turnaround times are negligible, and the boat that com- pletes the round trip first wins. (a) Which boat wins and by how much? (Or is it a tie?) (b) What is the average ve- locity of the winning boat?

The Olympic record for the marathon is 2 h, 9 min, 21 s. The marathon distance is 26 mi, 385 yd. Determine the average speed (in miles per hour) of the record.

A motorist drives north for 35.0 minutes at 85.0 km/h and then stops for 15.0 minutes. He then continues north, traveling 130 km in 2.00 h. (a) What is his total dis- placement? (b) What is his average velocity?

A graph of position versus time for a certain particle moving along the x-axis is shown in Figure P2.6. Find the average velocity in the time intervals from (a) 0 to 2.00 s, (b) 0 to 4.00 s, (c) 2.00 s to 4.00 s, (d) 4.00 s to 7.00 s, and (e) 0 to 8.00 s.

A tennis player moves in a straight-line path as shown in Figure P2.7. Find her average velocity in the time in- tervals from (a) 0 to 1.0 s, (b) 0 to 4.0 s, (c) 1.0 s to 5.0 s, and (d) 0 to 5.0

Two cars travel in the same direction along a straight highway, one at a constant speed of 55 mi/h and the other at 70 mi/h. (a) Assuming that they start at the same point, how much sooner does the faster car arrive at a des- tination 10 mi away? (b) How far must the faster car travel before it has a 15-min lead on the slower car?

An athlete swims the length of a 50.0-m pool in 20.0 s and makes the return trip to the starting position in 22.0 s. Determine her average velocities in (a) the first half of the swim, (b) the second half of the swim, and (c) the round trip.

If the average speed of an orbiting space shuttle is 19 800 mi/h, determine the time required for it to circle Earth. Make sure you consider the fact that the shuttle is orbiting about 200 mi above Earths surface, and assume that Earths radius is 3 963 miles.

A person takes a trip, driving with a con- stant speed of 89.5 km/h, except for a 22.0-min rest stop. If the persons average speed is 77.8 km/h, how much time is spent on the trip and how far does the person travel?

A tortoise can run with a speed of 0.10 m/s, and a hare can run 20 times as fast. In a race, they both start at the same time, but the hare stops to rest for 2.0 minutes. The tortoise wins by a shell (20 cm). (a) How long does the race take? (b) What is the length of the race?

In order to qualify for the finals in a racing event, a race car must achieve an average speed of 250 km/h on a track with a total length of 1 600 m. If a particular car covers the first half of the track at an average speed of 230 km/h, what minimum average speed must it have in the second half of the event in order to qualify?

Runner A is initially 4.0 mi west of a flagpole and is run- ning with a constant velocity of 6.0 mi/h due east. Runner B is initially 3.0 mi east of the flagpole and is running with a constant velocity of 5.0 mi/h due west. How far are the runners from the flagpole when they meet?

A graph of position versus time for a certain particle moving along the x-axis is shown in Figure P2.6. Find the instantaneous velocity at the instants (a) t 1.00 s, (b) t 3.00 s, (c) t 4.50 s, and (d) t 7.50 s.

A race car moves such that its position fits the relationship x (5.0 m/s)t (0.75 m/s3)t 3 where x is measured in meters and t in seconds. (a) Plot a graph of the cars position versus time. (b) Determine the instantaneous velocity of the car at t 4.0 s, using time intervals of 0.40 s, 0.20 s, and 0.10 s. (c) Compare the average velocity during the first 4.0 s with the results of (b).

Find the instantaneous velocities of the tennis player of Figure P2.7 at (a) 0.50 s, (b) 2.0 s, (c) 3.0 s, and (d) 4.5 s.

Secretariat ran the Kentucky Derby with times of 25.2 s, 24.0 s, 23.8 s, and 23.0 s for the quarter mile. (a) Find his average speed during each quarter-mile segment. (b) Assuming that Secretariats instantaneous speed at the finish line was the same as his average speed during the final quarter mile, find his average acceleration for the entire race.

A steam catapult launches a jet aircraft from the aircraft carrier John C. Stennis, giving it a speed of 175 mi/h in 2.50 s. (a) Find the average acceleration of the plane. (b) Assuming that the acceleration is constant, find the distance the plane moves.

A car traveling in a straight line has a velocity of 5.0 m/s at some instant. After 4.0 s, its velocity is 8.0 m/s. What is the cars

A certain car is capable of accelerating at a rate of 0.60 m/s2. How long does it take for this car to go from a speed of 55 mi/h to a speed of 60 mi/h?

The velocity vs. time graph for an object moving along a straight path is shown in Figure P2.22. (a) Find the aver- age acceleration of the object during the time intervals 0 to 5.0 s, 5.0 s to 15 s, and 0 to 20 s. (b) Find the instanta- neous acceleration at 2.0 s, 10 s, and 18 s.

The engine of a model rocket accelerates the rocket verti- cally upward for 2.0 s as follows: At t 0, the rockets speed is zero; at t 1.0 s, its speed is 5.0 m/s; and at t 2.0 s, its speed is 16 m/s. Plot a velocity vs. time graph for this motion, and use the graph to determine (a) the rockets average acceleration during the 2.0-s interval and (b) the instantaneous acceleration of the rocket at t 1.5 s.

A car traveling in a straight-line path has a velocity of 10.0 m/s at some instant. After 3.00 s, its velocity is 6.00 m/s. What is the average acceleration of the car during this time interval?

In 1865, Jules Verne proposed sending men to the Moon by firing a space capsule from a 220-m-long cannon with final speed of 10.97 km/s. What would have been the un- realistically large acceleration experienced by the space travelers during their launch? (A human can stand an ac- celeration of 15g for a short time.) Compare your answer with the free-fall acceleration, 9.80 m/s2.

A truck covers 40.0 m in 8.50 s while smoothly slowing down to a final speed of 2.80 m/s. (a) Find the trucks original speed. (b) Find its acceleration.

A speedboat increases its speed uniformly from 20 m/s to 30 m/s in a distance of 200 m. Find (a) the magnitude of its acceleration and (b) the time it takes the boat to travel the 200-m distance.

Two cars are traveling along a straight line in the same direction, the lead car at 25.0 m/s and the other car at 30.0 m/s. At the moment the cars are 40.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of 2.00 m/s2. (a) How long does it take for the lead car to stop? (b) Assuming that the chasing car brakes at the same time as the lead car, what must be the chasing cars minimum negative acceleration so as not to hit the lead car? (c) How long does it take for the chasing car to stop?

A Cessna aircraft has a lift-off speed of 120 km/h. (a) What minimum constant acceleration does the air- craft require if it is to be airborne after a takeoff run of 240 m? (b) How long does it take the aircraft to become airborne?

A truck on a straight road starts from rest and accelerates at 2.0 m/s2 until it reaches a speed of 20 m/s. Then the truck travels for 20 s at constant speed until the brakes are applied, stopping the truck in a uniform manner in an additional 5.0 s. (a) How long is the truck in motion? (b) What is the average velocity of the truck during the motion described?

A drag racer starts her car from rest and accelerates at 10.0 m/s2 for a distance of 400 m ( mile). (a) How long did it take the race car to travel this distance? (b) What is the speed of the race car at the end of the run?

A jet plane lands with a speed of 100 m/s and can acceler- ate at a maximum rate of 5.00 m/s2 as it comes to rest. (a) From the instant the plane touches the runway, what is the minimum time needed before it can come to rest? (b) Can this plane land on a small tropical island airport where the runway is 0.800 km long?

A driver in a car traveling at a speed of 60 mi/h sees a deer 100 m away on the road. Calculate the minimum constant acceleration that is necessary for the car to stop without hitting the deer (assuming that the deer does not move in the meantime).

A record of travel along a straight path is as follows: 1. Start from rest with a constant acceleration of 2.77 m/s2 for 15.0 s. 2. Maintain a constant velocity for the next 2.05 min. 3. Apply a constant negative acceleration of 9.47 m/s2 for 4.39 s. (a) What was the total displacement for the trip? (b) What were the average speeds for legs 1, 2, and 3 of the trip, as well as for the complete trip?

A train is traveling down a straight track at 20 m/s when the engineer applies the brakes, resulting in an accelera- tion of 1.0 m/s2 as long as the train is in motion. How far does the train move during a 40-s time interval start- ing at the instant the brakes are applied?

A car accelerates uniformly from rest to a speed of 40.0 mi/h in 12.0 s. Find (a) the distance the car travels dur- ing this time and (b) the constant acceleration of the car.

A car starts from rest and travels for 5.0 s with a uniform acceleration of 1.5 m/s2. The driver then applies the brakes, causing a uniform acceleration of 2.0 m/s2. If the brakes are applied for 3.0 s, (a) how fast is the car go- ing at the end of the braking period, and (b) how far has the car gone?

A train 400 m long is moving on a straight track with a speed of 82.4 km/h. The engineer applies the brakes at a crossing, and later the last car passes the crossing with a speed of 16.4 km/h. Assuming constant acceleration, de- termine how long the train blocked the crossing. Disre- gard the width of the crossing.

A hockey player is standing on his skates on a frozen pond when an opposing player, moving with a uniform speed of 12 m/s, skates by with the puck. After 3.0 s, the first player makes up his mind to chase his opponent. If he acceler- ates uniformly at 4.0 m/s2, (a) how long does it take him to catch his opponent, and (b) how far has he traveled in that time? (Assume that the player with the puck remains in motion at constant speed.)

A glider on an air track carries a flag of length through a stationary photogate that measures the time interval td during which the flag blocks a beam of infrared light pass- ing across the gate. The ratio vd /td is the average ve- locity of the glider over this part of its motion. Suppose the glider moves with constant acceleration. (a) Argue for or against the idea that vd is equal to the instantaneous velocity of the glider when it is halfway through the photogate in terms of distance. (b) Argue for or against the idea that vd is equal to the instantaneous velocity of the glider when it is halfway through the photogate in terms of time.

In the Daytona 500 auto race, a Ford Thunderbird and a Mercedes Benz are moving side by side down a straight- away at 71.5 m/s. The driver of the Thunderbird realizes that she must make a pit stop, and she smoothly slows to a stop over a distance of 250 m. She spends 5.00 s in the pit and then accelerates out, reaching her previous speed of 71.5 m/s after a distance of 350 m. At this point, how far has the Thunderbird fallen behind the Mercedes Benz, which has continued at a constant speed?

A certain cable car in San Francisco can stop in 10 s when traveling at maximum speed. On one occasion, the driver sees a dog a distance d m in front of the car and slams on the brakes instantly. The car reaches the dog 8.0 s later, and the dog jumps off the track just in time. If the car travels 4.0 m beyond the position of the dog before coming to a stop, how far was the car from the dog? (Hint: You will need three equations.)

A ball is thrown vertically upward with a speed of 25.0 m/s. (a) How high does it rise? (b) How long does it take to reach its highest point? (c) How long does the ball take to hit the ground after it reaches its highest point? (d) What is its velocity when it returns to the level from which it started?

It is possible to shoot an arrow at a speed as high as 100 m/s. (a) If friction is neglected, how high would an arrow launched at this speed rise if shot straight up? (b) How long would the arrow be in the air?

A certain freely falling object requires 1.50 s to travel the last 30.0 m before it hits the ground. From what height above the ground did it fall?

Traumatic brain injury such as concussion results when the head undergoes a very large acceleration. Generally, an acceleration less than 800 m/s2 lasting for any length of time will not cause injury, whereas an acceleration greater than 1 000 m/s2 lasting for at least 1 ms will cause injury. Suppose a small child rolls off a bed that is 0.40 m above the floor. If the floor is hardwood, the childs head is brought to rest in approximately 2.0 mm. If the floor is carpeted, this stopping distance is increased to about 1.0 cm. Calculate the magnitude and duration of the de- celeration in both cases, to determine the risk of injury. Assume that the child remains horizontal during the fall to the floor. Note that a more complicated fall could result in a head velocity greater or less than the speed you calculate.

A small mailbag is released from a heli- copter that is descending steadily at 1.50 m/s. After 2.00 s, (a) what is the speed of the mailbag, and (b) how far is it below the helicopter? (c) What are your answers to parts (a) and (b) if the helicopter is rising steadily at 1.50 m/s?

A ball thrown vertically upward is caught by the thrower after 2.00 s. Find (a) the initial velocity of the ball and (b) the maximum height the ball reaches.

A model rocket is launched straight upward with an initial speed of 50.0 m/s. It accelerates with a constant upward acceleration of 2.00 m/s2 until its engines stop at an alti- tude of 150 m. (a) What is the maximum height reached by the rocket? (b) How long after lift-off does the rocket reach its maximum height? (c) How long is the rocket in the air?

A parachutist with a camera descends in free fall at a speed of 10 m/s. The parachutist releases the camera at an altitude of 50 m. (a) How long does it take the camera to reach the ground? (b) What is the

A student throws a set of keys vertically upward to his fraternity brother, who is in a window 4.00 m above. The brothers outstretched hand catches the keys 1.50 s later. (a) With what initial velocity were the keys thrown? (b) What was the velocity of the keys just before they were caught?

It has been claimed that an insect called the froghopper (Philaenus spumarius) is the best jumper in the animal kingdom. This insect can accelerate at 4 000 m/s2 over a distance of 2.0 mm as it straightens its specially designed jumping legs. (a) Assuming a uniform acceleration, what is the velocity of the insect after it has accelerated through this short distance, and how long did it take to reach that velocity? (b) How high would the insect jump if air resistance could be ignored? Note that the actual height obtained is about 0.7 m, so air resistance is impor- tant here.

A truck tractor pulls two trailers, one behind the other, at a constant speed of 100 km/h. It takes 0.600 s for the big rig to completely pass onto a bridge 400 m long. For what duration of time is all or part of the truck-trailer combina- tion on the bridge?

A speedboat moving at 30.0 m/s approaches a no-wake buoy marker 100 m ahead. The pilot slows the boat with a constant acceleration of 3.50 m/s2 by reducing the throttle. (a) How long does it take the boat to reach the buoy? (b) What is the velocity of the boat when it reaches the buoy?

A bullet is fired through a board 10.0 cm thick in such a way that the bullets line of motion is perpendicular to the face of the board. If the initial speed of the bullet is 400 m/s and it emerges from the other side of the board with a speed of 300 m/s, find (a) the acceleration of the bullet as it passes through the board and (b) the total time the bullet is in contact with the board.

An indestructible bullet 2.00 cm long is fired straight through a board that is 10.0 cm thick. The bullet strikes the board with a speed of 420 m/s and emerges with a speed of 280 m/s. (a) What is the average acceleration of the bullet through the board? (b) What is the total time that the bullet is in contact with the board? (c) What thickness of board (calculated to 0.1 cm) would it take to stop the bullet, assuming that the acceleration through all boards is the same?

A ball is thrown upward from the ground with an initial speed of 25 m/s; at the same instant, another ball is dropped from a building 15 m high. After how long will the balls be at the same height?

A ranger in a national park is driving at 35.0 mi/h when a deer jumps into the road 200 ft ahead of the vehicle. After a reaction time t, the ranger applies the brakes to produce an acceleration a 9.00 ft/s2. What is the maximum re- action time allowed if she is to avoid hitting the deer?

Two students are on a balcony 19.6 m above the street. One student throws a ball vertically downward at 14.7 m/s; at the same instant, the other student throws a ball vertically upward at the same speed. The second ball just misses the balcony on the way down. (a) What is the difference in the two balls time in the air? (b) What is the velocity of each ball as it strikes the ground? (c) How far apart are the balls 0.800 s after they are thrown?

The driver of a truck slams on the brakes when he sees a tree blocking the road. The truck slows down uniformly with an acceleration of 5.60 m/s2 for 4.20 s, making skid marks 62.4 m long that end at the tree. With what speed does the truck then strike the tree?

A young woman named Kathy Kool buys a sports car that can accelerate at the rate of 4.90 m/s2. She decides to test the car by drag racing with another speedster, Stan Speedy. Both start from rest, but experienced Stan leaves the starting line 1.00 s before Kathy. If Stan moves with a constant acceleration of 3.50 m/s2 and Kathy maintains an acceleration of 4.90 m/s2, find (a) the time it takes Kathy to overtake Stan, (b) the distance she travels before she catches him, and (c) the speeds of both cars at the in- stant she overtakes him.

A mountain climber stands at the top of a 50.0-m cliff that overhangs a calm pool of water. She throws two stones ver- tically downward 1.00 s apart and observes that they cause a single splash. The first stone had an initial velocity of 2.00 m/s. (a) How long after release of the first stone did the two stones hit the water? (b) What initial velocity must the second stone have had, given that they hit the water simultaneously? (c) What was the velocity of each stone at the instant it hit the water?

An ice sled powered by a rocket engine starts from rest on a large frozen lake and accelerates at 40 ft/s2. After some time t1, the rocket engine is shut down and the sled moves with constant velocity v for a time t 2. If the total distance traveled by the sled is 17 500 ft and the total time is 90 s, find (a) the times t1 and t 2 and (b) the velocity v. At the 17 500-ft mark, the sled begins to accelerate at 20 ft/s2. (c) What is the final position of the sled when it comes to rest? (d) How long does it take to come to rest?

In Bosnia, the ultimate test of a young mans courage used to be to jump off a 400-year-old bridge (now de- stroyed) into the River Neretva, 23 m below the bridge. (a) How long did the jump last? (b) How fast was the jumper traveling upon impact with the river? (c) If the speed of sound in air is 340 m/s, how long after the jumper took off did a spectator on the bridge hear the splash?

A person sees a lightning bolt pass close to an airplane that is flying in the distance. The person hears thunder 5.0 s after seeing the bolt and sees the airplane overhead 10 s after hearing the thunder. The speed of sound in air is 1 100 ft/s. (a) Find the distance of the airplane from the person at the instant of the bolt. (Neglect the time it takes the light to travel from the bolt to the eye.) (b) As- suming that the plane travels with a constant speed toward the person, find the velocity of the airplane. (c) Look up the speed of light in air, and defend the ap- proximation used in (a).

Another scheme to catch the roadrunner has failed! Now a safe falls from rest from the top of a 25.0-m-high cliff to- ward Wile E. Coyote, who is standing at the base. Wile first notices the safe after it has fallen 15.0 m. How long does he have to get out of the way?

A stunt man sitting on a tree limb wishes to drop vertically onto a horse galloping under the tree. The constant speed of the horse is 10.0 m/s, and the man is initially 3.00 m above the level of the saddle. (a) What must be the horizontal distance between the saddle and the limb when the man makes his move? (b) How long is he in the air?

A hard rubber ball, released at chest height, falls to the pavement and bounces back to nearly the same height. When the ball is in contact with the pavement, its lower side is temporarily flattened. Before the dent in the ball pops out, suppose that its maximum depth is on the order of 1 cm. Compute an order-of-magnitude estimate for the maximum acceleration of the ball. State your assump- tions, the quantities you estimate, and the values you esti- mate for them.

Vroomvroom! As soon as a traffic light turns green, a car speeds up from rest to 50.0 mi/h with a constant accelera- tion of 9.00 mi/hs. In the adjoining bike lane, a cyclist speeds up from rest to 20.0 mi/h with a constant accelera- tion of 13.0 mi/hs. Each vehicle maintains a constant velocity after reaching its cruising speed. (a) For how long is the bicycle ahead of the car? (b) By what maximum dis- tance does the bicycle lead the car?

In order to pass a physical education class at a university, a student must run 1.0 mi in 12 min. After running for 10 min, she still has 500 yd to go. If her maximum acceleration is 0.15 m/s2, can she make it? If the answer is no, determine what acceleration she would need to be successful.

One swimmer in a relay race has a 0.50-s lead and is swim- ming at a constant speed of 4.0 m/s. He has 50 m to swim before reaching the end of the pool. A second swimmer moves in the same direction as the leader. What constant speed must the second swimmer have in order to catch up to the leader at the end of the pool?

Estimate a few speeds in metric units, using a stopwatch or a wristwatch. For example, roll a ball across a table and estimate the number of centimeters it moves each second to find its speed. Other speeds you might try are for some- one walking across the room, a jogger running, a car mov- ing through some distance, and so forth. To see how well you did, make some actual measurements for those situa- tions in which it is feasible to do so.

Use what you know about falling objects to measure your reaction time. Hold the index finger and thumb of your dominant hand about 2.5 cm apart, and then have your co-worker hold a ruler vertically in the space between your finger and thumb, as shown in Figure A2.2. Note the position of the ruler relative to your index finger. Your co-worker must release the ruler, and you must catch it (without moving your hand downward) as quickly as you can. The ruler (a freely falling object) falls through a distance , where d t is the reaction time and 1 Repeat this measurement of d five times, average your results, and calculate an average value of t. Now measure your co-workers reaction time, using the same procedure. Compare your results. For most people, the reaction time is at best about 0.2 s. As an extension to this experiment, replace the ruler with a crisp dollar bill. Hold the bill such that your thumb and index finger are just at the level of Washingtons face. Unless you are antic- ipating the time of release, you will not be able to catch the bill when it is released, because the time required for the top to pass out of your hand is less than the typical 0.2-s reaction time.

Galileo studied accelerated motion by allowing objects to roll down inclined planes so that their motion would be slow enough to make reasonable observations. Try a simi- lar procedure. Make a mark at the top of an inclined plane as the starting point for the motion, and use a metal barrier at the end as a sound cue for stopping a stop- watch. Measure the length of the plane. Record the aver- age time for several trials of a ball rolling down the plane at a measured angle. From the information you obtain, calculate the acceleration. Repeat the experiment for a larger angle of inclination. Do this for several trials, until you can plot a graph of acceleration versus angle. From your graph, can you guess what the acceleration would be if the inclined plane were vertical? Would the results of your experiment be different if you had used a signifi- cantly more massive ball? If you are unsure, repeat the ex- periment to see if there is a difference.

Perform the activities that follow to verify that all objects fall with the same acceleration. First, try dropping a coffee filter oriented horizontally and also dropping a pencil. Then repeat the experiment with the filter in a loose ball, a tight ball, and, finally, in a compacted wad. You should note that compacting the filter tends to reduce the effects of air resistance and makes the two objects fall more nearly at the same rate.