Solved: Consider two possible definitions of average speed: (a) the average of the

Under what conditions are average and instantaneous velocity equal?

Does a speedometer measure speed or velocity?

You check your odometer at the beginning of a days driving and again at the end. Under what conditions would the difference between the two readings represent your displacement?

Consider two possible definitions of average speed: (a) the average of the values of the instantaneous speed over a time interval and (b) the magnitude of the average velocity. Are these definitions equivalent? Give two examples to demonstrate your conclusion.

Is it possible to be at position x = 0 and still be moving?

Is it possible to have zero velocity and still be accelerating?

If you know the initial velocity v0 and the initial and final heights y0 and y, you can use Equation 2.10 to solve for the time t when the object will be at height y. But the equation is quadratic in t, so youll get two answers. Physically, why is this?

Starting from rest, an object undergoes acceleration given by a = bt, where t is time and b is a constant. Can you use bt for a in Equation 2.10 to predict the objects position as a function of time? Why or why not?

In which of the velocity-versus-time graphs shown in Fig. 2.14 would the average velocity over the interval shown equal the average of the velocities at the ends of the interval?

If you travel in a straight line at 50 km/h for 1 h and at 100 km/h for another hour, is your average velocity 75 km/h? If not, is it more or less?

If you travel in a straight line at 50 km/h for 50 km and then at 100 km/h for another 50 km, is your average velocity 75 km/h? If not, is it more or less?

In 2009, Usain Bolt of Jamaica set a world record in the 100-m dash with a time of 9.58 s. What was his average speed?

The standard 26-mile, 385-yard marathon dates to 1908, when the Olympic marathon started at Windsor Castle and finished before the Royal Box at Londons Olympic Stadium. Todays top marathoners achieve times around 2 hours, 3 minutes for the standard marathon. (a) Whats the average speed of a marathon run in this time? (b) Marathons before 1908 were typically about 25 miles. How much longer does the race last today as a result of the extra mile and 385 yards, assuming its run at part (a)s average speed?

Starting from home, you bicycle 24 km north in 2.5 h and then turn around and pedal straight home in 1.5 h. What are your (a) displacement at the end of the first 2.5 h, (b) average velocity over the first 2.5 h, (c) average velocity for the homeward leg of the trip, (d) displacement for the entire trip, and (e) average velocity for the entire trip?

The Voyager 1 spacecraft is expected to continue broadcasting data until at least 2020, when it will be some 14 billion miles from Earth. How long will it take Voyagers radio signals, traveling at the speed of light, to reach Earth from this distance?

In 2008, Australian Emma Snowsill set an unofficial record in the womens Olympic triathlon, completing the 1.5-km swim, 40-km bicycle ride, and 10-km run in 1 h, 58 min, 27.66 s. What was her average speed?

Taking Earths orbit to be a circle of radius 1.5 * 108 km, determine Earths orbital speed in (a) meters per second and (b) miles per second.

Whats the conversion factor from meters per second to miles per hour?

On a single graph, plot distance versus time for the first two trips from Houston to Des Moines described on page 16. For each trip, identify graphically the average velocity and, for each segment of the trip, the instantaneous velocity

For the motion plotted in Fig. 2.15, estimate (a) the greatest velocity in the positive x-direction, (b) the greatest velocity in the negative x-direction, (c) any times when the object is instantaneously at rest, and (d) the average velocity over the interval shown.

A model rocket is launched straight upward. Its altitude y as a function of time is given by y = bt - ct2 , where b = 82 m/s, c = 4.9 m/s2 , t is the time in seconds, and y is in meters. (a) Use differentiation to find a general expression for the rockets velocity as a function of time. (b) When is the velocity zero?

A giant eruption on the Sun propels solar material from rest to 450 km/s over a period of 1 h. Find the average acceleration.

Starting from rest, a subway train first accelerates to 25 m/s, then brakes. Forty-eight seconds after starting, its moving at 17 m/s. Whats its average acceleration in this 48-s interval?

A space shuttles main engines cut off 8.5 min after launch, at which time its speed is 7.6 km/s. Whats the shuttles average acceleration during this interval?

An egg drops from a second-story window, taking 1.12 s to fall and reaching 11.0 m/s just before hitting the ground. On contact, the egg stops completely in 0.131 s. Calculate the magnitudes of its average acceleration (a) while falling and (b) while stopping.

An airplanes takeoff speed is 320 km/h. If its average acceleration is 2.9 m/s2 , how much time is it accelerating down the runway before it lifts off?

ThrustSSC, the worlds first supersonic car, accelerates from rest to 1000 km/h in 16 s. Whats its acceleration?

Youre driving at 70 km/h when you apply constant acceleration to pass another car. Six seconds later, youre doing 80 km/h. How far did you go in this time?

Differentiate both sides of Equation 2.10, and show that you get Equation 2.7

An X-ray tube gives electrons constant acceleration over a distance of 15 cm. If their final speed is 1.2 * 107 m/s, what are (a) the electrons acceleration and (b) the time they spend accelerating?

A rocket rises with constant acceleration to an altitude of 85 km, at which point its speed is 2.8 km/s. (a) Whats its acceleration? (b) How long does the ascent take?

Starting from rest, a car accelerates at a constant rate, reaching 88 km/h in 12 s. Find (a) its acceleration and (b) how far it goes in this time.

A car moving initially at 50 mi/h begins slowing at a constant rate 100 ft short of a stoplight. If the car comes to a full stop just at the light, what is the magnitude of its acceleration?

In a medical X-ray tube, electrons are accelerated to a velocity of 108 m/s and then slammed into a tungsten target. As they stop, the electrons rapid acceleration produces X rays. If the time for an electron to stop is on the order of 10-9 s, approximately how far does it move while stopping?

Californias Bay Area Rapid Transit System (BART) uses an automatic braking system triggered by earthquake warnings. The system is designed to prevent disastrous accidents involving trains traveling at a maximum of 112 km/h and carrying a total of some 45,000 passengers at rush hour. If it takes a train 24 s to brake to a stop, how much advance warning of an earthquake is needed to bring a 112-km/h train to a reasonably safe speed of 42 km/h when the earthquake strikes?

Youre driving at speed v0 when you spot a stationary moose on the road, a distance d ahead. Find an expression for the magnitude of the acceleration you need if youre to stop before hitting the moose.

You drop a rock into a deep well and 4.4 s later hear a splash. How far down is the water? Neglect the travel time of sound.

Your friend is sitting 6.5 m above you on a tree branch. How fast should you throw an apple so it just reaches her?

A model rocket leaves the ground, heading straight up at 49 m/s. (a) Whats its maximum altitude? Find its speed and altitude at (b) 1 s, (c) 4 s, and (d) 7 s.

A foul ball leaves the bat going straight up at 23 m/s. (a) How high does it rise? (b) How long is it in the air? Neglect the distance between bat and ground.

A Frisbee is lodged in a tree 6.5 m above the ground. A rock thrown from below must be going at least 3 m/s to dislodge the Frisbee. How fast must such a rock be thrown upward if it leaves the throwers hand 1.3 m above the ground?

Space pirates kidnap an earthling and hold him on one of the solar systems planets. With nothing else to do, the prisoner amuses himself by dropping his watch from eye level (170 cm) to the floor. He observes that the watch takes 0.95 s to fall. On what planet is he being held? (Hint: Consult Appendix E.)

You allow 40 min to drive 25 mi to the airport, but youre caught in heavy traffic and average only 20 mi/h for the first 15 min. What must your average speed be on the rest of the trip if youre to make your flight?

A base runner can get from first to second base in 3.4 s. If he leaves first as the pitcher throws a 90 mi/h fastball the 61-ft distance to the catcher, and if the catcher takes 0.45 s to catch and rethrow the ball, how fast does the catcher have to throw the ball to second base to make an out? Home plate to second base is the diagonal of a square 90 ft on a side.

You can run 9.0 m/s, 20% faster than your brother. How much head start should you give him in order to have a tie race over 100 m?

A jetliner leaves San Francisco for New York, 4600 km away. With a strong tailwind, its speed is 1100 km/h. At the same time, a second jet leaves New York for San Francisco. Flying into the wind, it makes only 700 km/h. When and where do the two planes pass?

An objects position is given by x = bt + ct3 , where b = 1.50 m/s, c = 0.640 m/s3 , and t is time in seconds. To study the limiting process leading to the instantaneous velocity, calculate the objects average velocity over time intervals from (a) 1.00 s to 3.00 s, (b) 1.50 s to 2.50 s, and (c) 1.95 s to 2.05 s. (d) Find the instantaneous velocity as a function of time by differentiating, and compare its value at 2 s with your average velocities

An objects position as a function of time t is given by x = bt4 , with b a constant. Find an expression for the instantaneous velocity, and show that the average velocity over the interval from t = 0 to any time t is one-fourth of the instantaneous velocity at t.

In a drag race, the position of a car as a function of time is given by x = bt2 , with b = 2.000 m/s2 . In an attempt to determine the cars velocity midway down a 400-m track, two observers stand at the 180-m and 220-m marks and note when the car passes. (a) What value do the two observers compute for the cars velocity over this 40-m stretch? Give your answer to four significant figures. (b) By what percentage does this observed value differ from the instantaneous value at x = 200 m?

Squaring Equation 2.7 gives an expression for v2 . Equation 2.11 also gives an expression for v2 . Equate the two expressions, and show that the resulting equation reduces to Equation 2.10.

During the complicated sequence that landed the rover Curiosity on Mars in 2012, the spacecraft reached an altitude of 142 m above the Martian surface, moving vertically downward at 32.0 m/s. It then entered a so-called constant deceleration (CD) phase, during which its velocity decreased steadily to 0.75 m/s while it dropped to an altitude of 23 m. What was the magnitude of the spacecrafts acceleration during this CD phase?

The position of a car in a drag race is measured each second, and the results are tabulated below. Time t (s) 0 1 2 3 4 5 Position x (m) 0 1.7 6.2 17 24 40 Assuming the acceleration is approximately constant, plot position versus a quantity that should make the graph a straight line. Fit a line to the data, and from it determine the approximate acceleration

A fireworks rocket explodes at a height of 82.0 m, producing fragments with velocities ranging from 7.68 m/s downward to 16.7 m/s upward. Over what time interval are fragments hitting the ground?

The muscles in a grasshoppers legs can propel the insect upward at 3.0 m/s. How high can the grasshopper jump?

On packed snow, computerized antilock brakes can reduce a cars stopping distance by 55%. By what percentage is the stopping time reduced?

A particle leaves its initial position x0 at time t = 0, moving in the positive x-direction with speed v0 but undergoing acceleration of magnitude a in the negative x-direction. Find expressions for (a) the time when it returns to x0 and (b) its speed when it passes that point

A hockey puck moving at 32 m/s slams through a wall of snow 35 cm thick. It emerges moving at 18 m/s. Assuming constant acceleration, find (a) the time the puck spends in the snow and (b) the thickness of a snow wall that would stop the puck entirely.

Amtraks 20th-Century Limited is en route from Chicago to New York at 110 km/h when the engineer spots a cow on the track. The train brakes to a halt in 1.2 min, stopping just in front of the cow. (a) What is the magnitude of the trains acceleration? (b) Whats the direction of the acceleration? (c) How far was the train from the cow when the engineer applied the brakes?

A jetliner touches down at 220 km/h and comes to a halt 29 s later. Whats the shortest runway on which this aircraft can land?

A motorist suddenly notices a stalled car and slams on the brakes, negatively accelerating at 6.3 m/s2 . Unfortunately, this isnt enough, and a collision ensues. From the damage sustained, police estimate that the car was going 18 km/h at the time of the collision. They also measure skid marks 34 m long. (a) How fast was the motorist going when the brakes were first applied? (b) How much time elapsed from the initial braking to the collision?

A racing car undergoing constant acceleration covers 140 m in 3.6 s. (a) If its moving at 53 m/s at the end of this interval, what was its speed at the beginning of the interval? (b) How far did it travel from rest to the end of the 140-m distance?

The maximum braking acceleration of a car on a dry road is about 8 m/s2 . If two cars move head-on toward each other at 88 km/h (55 mi/h), and their drivers brake when theyre 85 m apart, will they collide? If so, at what relative speed? If not, how far apart will they be when they stop? Plot distance versus time for both cars on a single graph.

After 35 min of running, at the 9-km point in a 10-km race, you find yourself 100 m behind the leader and moving at the same speed. What should your acceleration be if youre to catch up by the finish line? Assume that the leader maintains constant speed.

Youre speeding at 85 km/h when you notice that youre only 10 m behind the car in front of you, which is moving at the legal speed limit of 60 km/h. You slam on your brakes, and your car negatively accelerates at 4.2 m/s2 . Assuming the other car continues at constant speed, will you collide? If so, at what relative speed? If not, what will be the distance between the cars at their closest approach?

Airbags cushioned the Mars rover Spirits landing, and the rover bounced some 15 m vertically after its first impact. Assuming no loss of speed at contact with the Martian surface, what was Spirits impact speed?

Calculate the speed with which cesium atoms must be tossed in the NIST-F1 atomic clock so that their up-and-down travel time is 1.0 s. (See the Application on page 26.)

A falling object travels one-fourth of its total distance in the last second of its fall. From what height was it dropped?

Youre on a NASA team engineering a probe to land on Jupiters moon Io, and your job is to specify the impact speed the probe can tolerate without damage. Rockets will bring the probe to a halt 100 m above the surface, after which it will fall freely. What speed do you specify? (Consult Appendix E.)

Youre atop a building of height h, and a friend is poised to drop a ball from a window at h/2. Find an expression for the speed at which you should simultaneously throw a ball downward, so the two hit the ground at the same time.

A castles defenders throw rocks down on their attackers from a 15-m-high wall, with initial speed 10 m/s. How much faster are the rocks moving when they hit the ground than if they were simply dropped?

Two divers jump from a 3.00-m platform. One jumps upward at 1.80 m/s, and the second steps off the platform as the first passes it on the way down. (a) What are their speeds as they hit the water? (b) Which hits the water first and by how much?

A balloon is rising at 10 m/s when its passenger throws a ball straight up at 12 m/s relative to the balloon. How much later does the passenger catch the ball?

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

Youre at mission control for a rocket launch, deciding whether to let the launch proceed. A band of clouds 5.3 km thick extends upward from 1.9 km altitude. The rocket will accelerate at 4.6 m/s2 , and it isnt allowed to be out of sight for more than 30 s. Should you allow the launch?

Youre an investigator for the National Transportation Safety Board, examining a subway accident in which a train going at 80 km/h collided with a slower train traveling in the same direction at 25 km/h. Your job is to determine the relative speed of the collision, to help establish new crash standards. The faster trains black box shows that its brakes were applied and it began slowing at the rate of 2.1 m/s2 when it was 50 m from the slower train, while the slower train continued at constant speed. What do you report?

You toss a book into your dorm room, just clearing a windowsill 4.2 m above the ground. (a) If the book leaves your hand 1.5 m above the ground, how fast must it be going to clear the sill? (b) How long after it leaves your hand will it hit the floor, 0.87 m below the windowsill?

Consider an object traversing a distance L, part of the way at speed v1 and the rest of the way at speed v2. Find expressions for the objects average speed over the entire distance L when the object moves at each of the two speeds v1 and v2 for (a) half the total time and (b) half the total distance. (c) In which case is the average speed greater?

A particles position as a function of time is given by x = x0 sinvt, where x0 and v are constants. (a) Find expressions for the velocity and acceleration. (b) What are the maximum values of velocity and acceleration? (Hint: Consult the table of derivatives in Appendix A.)

Ice skaters, ballet dancers, and basketball players executing vertical leaps often give the illusion of hanging almost motionless near the top of the leap. To see why this is, consider a leap to maximum height h. Of the total time spent in the air, what fraction is spent in the upper half (i.e., at y 7 1 2h)?

Youre staring idly out your dorm window when you see a water balloon fall past. If the balloon takes 0.22 s to cross the 1.3-m-high window, from what height above the window was it dropped?

A police radars effective range is 1.0 km, and your radar detectors range is 1.9 km. Youre going 110 km/h in a 70 km/h zone when the radar detector beeps. At what rate must you negatively accelerate to avoid a speeding ticket?

An object starts moving in a straight line from position x0, at time t = 0, with velocity v0. Its acceleration is given by a = a0 + bt, where a0 and b are constants. Use integration to find expressions for (a) the instantaneous velocity and (b) the position, as functions of time.

Youre a consultant on a movie set, and the producer wants a car to drop so that it crosses the cameras field of view in time t. The field of view has height h. Derive an expression for the height above the top of the field of view from which the car should be released.

(a) For the ball in Example 2.6, find its velocity just before it hits the floor. (b) Suppose you had tossed a second ball straight down at 7.3 m/s (from the same place 1.5 m above the floor). What would its velocity be just before it hits the floor? (c) When would the second ball hit the floor? (Interpret any multiple answers.)

Your roommate is an aspiring novelist and asks your opinion on a matter of physics. The novels central character is kept awake at night by a leaky faucet. The sink is 19.6 cm below the faucet. At the instant one drop leaves the faucet, another strikes the sink below and two more are in between on the way down. How many drops per second are keeping the protagonist awake?

You and your roommate plot to drop water balloons on students entering your dorm. Your window is 20 m above the sidewalk. You plan to place an X on the sidewalk to mark the spot a student must be when you drop the balloon. You note that most students approach the dorm at about 2 m/s. How far from the impact point do you place the X?

Derive Equation 2.10 by integrating Equation 2.7 over time. Youll have to interpret the constant of integration.

An objects acceleration increases quadratically with time: a(t) = bt2, where b = 0.041 m/s4. If the object starts from rest, how far does it travel in 6.3 s?

An objects acceleration is given by the expression a1t2 = -a0 cosvt, where a0 and v are positive constants. Find expressions for the objects (a) velocity and (b) position as functions of time. Assume that at time t = 0 it starts from rest at its greatest positive displacement from the origin. (c) Determine the magnitudes of the objects maximum velocity and maximum displacement from the origin

An objects acceleration decreases exponentially with time: a1t2 = a0e-bt, where a0 and b are constants. (a) Assuming the object starts from rest, determine its velocity as a function of time. (b) Will its speed increase indefinitely? (c) Will it travel indefinitely far from its starting point?

A ball is dropped from rest at a height h0 above the ground. At the same instant, a second ball is launched with speed v0 straight up from the ground, at a point directly below where the other ball is dropped. (a) Find a condition on v0 such that the two balls will collide in mid-air. (b) Find an expression for the height at which they collide.

A wildlife biologist is studying the hunting patterns of tigers. She anesthetizes a tiger and attaches a GPS collar to track its movements. The collar transmits data on the tigers position and velocity. Figure 2.16 shows the tigers velocity as a function of time as it moves on a one-dimensional path. At which marked point(s) is the tiger not moving? a. E only b. A, E, and H c. C and F d. none of the points (its always moving)

A wildlife biologist is studying the hunting patterns of tigers. She anesthetizes a tiger and attaches a GPS collar to track its movements. The collar transmits data on the tigers position and velocity. Figure 2.16 shows the tigers velocity as a function of time as it moves on a one-dimensional path. At which marked point(s) is the tiger not accelerating? a. E only b. A, E, and H c. C and F d. all of the points (its never accelerating)

A wildlife biologist is studying the hunting patterns of tigers. She anesthetizes a tiger and attaches a GPS collar to track its movements. The collar transmits data on the tigers position and velocity. Figure 2.16 shows the tigers velocity as a function of time as it moves on a one-dimensional path. At which point does the tiger have the greatest speed? a. B b. C c. D d. F

A wildlife biologist is studying the hunting patterns of tigers. She anesthetizes a tiger and attaches a GPS collar to track its movements. The collar transmits data on the tigers position and velocity. Figure 2.16 shows the tigers velocity as a function of time as it moves on a one-dimensional path. At which point does the tigers acceleration have the greatest magnitude? a. B b. C c. D d. F

A wildlife biologist is studying the hunting patterns of tigers. She anesthetizes a tiger and attaches a GPS collar to track its movements. The collar transmits data on the tigers position and velocity. Figure 2.16 shows the tigers velocity as a function of time as it moves on a one-dimensional path. At which point is the tiger farthest from its starting position at t = 0? a. C b. E c. F d. H