Cases – Chapter 1
1.You’re riding on a playground swing. You’re traveling back and forth once every few seconds.
a.At what point(s) in your motion is your velocity zero?
b.At what point(s) in your motion is your gravitational potential energy at its maximum?
c.At what point(s) in your motion is your kinetic energy at its maximum?
d.As you reach the bottom of a swing, when the swing’s ropes are exactly vertical, are you accelerating?
e.At the moment described in part d, is the force that the swing seat exerts on you more, less, or equal to your weight?
2.Diving boards and platforms offer a nearly ideal opportunity in which to experience the various laws of motion. When you jump off the high diving board, you are a falling object and, if you can keep your presence of mind as you fall, you can learn something about physics. Imagine yourself diving off a platform 10 m above the water below.
a.If you walk very slowly off the platform, so that you fall directly downward, roughly how long will it take for you to reach the water? Look at Fig. 1.2.2 and make a reasonable estimate.
b.In the situation described in part a, about how fast will you be traveling downward when you reach the water? Estimate your velocity from Fig. 1.2.2.
c.If you jump upward as you leave the platform, so that you begin with a modest upward velocity, will your downward velocity when you hit the water be more or less than in part b?
d.You leave the platform simultaneously with a friend. She walks slowly off the platform and you jump to give yourself a modest initial upward velocity. Who will reach the water first?
e.You leave the platform simultaneously with a friend. He walks slowly off the platform and you run off the platform, so that your initial velocity is in the horizontal direction. Who will reach the water first?
f.In the situation described in part e, who hits the water with the largest speed or are your speeds equal?
3.An escalator is essentially a moving staircase. The individual steps are supported by metal tracks that run on either side of the escalator. These steps follow one another in a complete loop, driven by an electric motor. When you step onto an escalator at the ground floor, it soon begins to carry you upward and forward at a constant velocity toward the second floor.
a.While you’re moving toward the second floor at a constant velocity, what is the net force exerted on you by all outside forces? (Specify the amount and the direction of the net force.)
b.You know that gravity gives you a weight in the downward direction. What force does the escalator exert on you as you move toward the second floor at a constant velocity? (Specify the amount and the direction of the force.)
c.Is the escalator doing work on you as you move toward the second floor?
d.As you first step onto the escalator, you begin to accelerate toward the second floor. Is the net force exerted on you by all outside forces the same as in part a?
e.If the rapidly moving escalator suddenly stopped moving, you would be thrown forward and might even fall over. What causes you to be thrown forward?
f.You have more energy when you reach the second floor than you had on the first floor. Why aren’t you moving faster as a result?
4.Imagine that you’re sledding alone down a steep hill on a toboggan and that you left the top of the hill at the same time as an identical toboggan, loaded with six people.
a.Neglecting the effects of air resistance and friction, which toboggan will reach the bottom of the hill first?
b.During the descent, your toboggan brushes up against the six-person toboggan. Which toboggan will experience the largest change in velocity as the result of the impact?
c.If you were to take a steeper route down the hill, how would that affect the speed of your descent? Explain.
d.Before each downhill run, you must pull the toboggan back to the top of the hill. Explain how the toboggan’s gravitational potential energy changes on the way up the hill and on the way down it.
e.When are you doing (positive) work on the toboggan?
f.When is gravity doing (positive) work on the toboggan?
5.You’re a pilot for the Navy. For your airplane to be able to lift itself off the ground, it must be traveling forward with a speed of 208 km/h (130 mph). At this takeoff speed your airplane will have about 50,000,000 N·m (or 50,000,000 J) of kinetic energy.
*a.During takeoff, your airplane’s jet engine exerts a force of 250,000 N in the backward direction on the air leaving the engine. What force does that same air exert on the airplane? (Specify the amount and the direction of force.)
*b.The force exerted by the air on the airplane causes it to accelerate down the runway. How long must the runway be for the airplane to reach its takeoff speed?
*c.An aircraft carrier runway is only about 100 m long. As your answer to part b indicates, this distance is not enough for the airplane to reach takeoff speed on its own. The aircraft carrier must assist the airplane by exerting an extra force on it. The aircraft carrier uses a steam-powered catapult to help push the airplane forward. How much additional force must the catapult exert on the airplane to bring the airplane to takeoff speed at the end of the 100-m runway?
d.During an aircraft carrier takeoff, the airplane and the catapult exert forces on one another. Which of these two objects does (positive) work on the other, and which object transfers some of its energy to the other?
e.During an aircraft carrier landing, the airplane hooks onto a cable that slows the airplane to a stop. The airplane and cable exert forces on one another. Which of these two objects does (positive) work on the other, and which object transfers some of its energy to the other?
6.You’re about to go on a bicycle trip through the mountains. Being ambitious, you decide to take two children along. The children sit in a trailer that you pull with your bicycle.
a.As you wait to begin your trip, you and your bicycle are motionless. What is the net force on your body?
b.You begin to bicycle into the mountains. You soon find yourself ascending a steep grade. The road rises smoothly uphill and you’re traveling up it at a steady pace in a straight line. You’re traveling at a constant velocity up the hill. What is the net force on your body?
*c.The road rises 1 m upward for every 10 m you travel along its surface. If the children and their trailer weigh 400 N, how much uphill force must you exert on the trailer to keep it moving uphill at a constant velocity?
*d.When you reach the top of the hill, your altitude has increased by 500 m. How much work did you do on the trailer and children as you pulled them up this hill? Does the amount of work you did on them depend on whether you took the long, gradually sloping road or the short, steep road? (Answer both questions.)
*e.How much work must you do on the trailer and children as you pull them back down the 500-m high hill at constant velocity?
7.You have recently taken up track and field as a way to keep in shape. You soon begin to notice how simple physical laws appear in many of the events.
a.You notice that great sprinters have extremely strong legs. Why is it so important that a sprinter be able to push back hard on the starting blocks at the beginning of a race?
b.You find that throwing a heavy metal shot is far more difficult than throwing a baseball. Weight isn’t the whole problem. Even if you try to throw the shot horizontally or downward, so that weight is not an issue, you have great difficulty getting the shot to move quickly. Why?
c.As you land on the soft foam pad beneath the pole vault, you realize that its job is to bring you to rest by accelerating you upward gradually with only modest support forces. If there were no pad there, only concrete, what would the acceleration and support forces be like during your landing?
d.You cross the finish line at the end of a race. The net force on your body points in what direction as you slow down?
e.In the long jump, you run rapidly down a path and then leap into the air. You find that the best distance comes from pushing yourself upward rather than forward during the leap. Why is it so important to have a large upward component of velocity at the start of the leap?
8.You’re juggling 3baseballs so that one of them is always in the air.
a.As you catch one of the falling balls, which way does the ball accelerate?
b.As you toss the ball back into the air, which way does the ball accelerate?
c.In both a and b, how does the upward force that you exert on the ball compare with the ball’s weight?
d.Even though one of the balls is always in the air, the average upward force you must exert on the balls is equal to their combined weight. Explain.
9.You have just started to coast down a hill on your bicycle. The hill has a very steady slope so that you descend 1m for every 5m of travel along the hill.
*a.Explain why the amount of force accelerating you and the bicycle downhill is only a fifth of your combined weights.
*b.How much does your velocity increase during the 1st second of coasting?
*c.How much does your velocity increase during the 2nd second of coasting?
d.Why do you travel farther during the 2nd second than during the 1st second of coasting?
10.Two cars are driving at 88km/h (55mph) toward an intersection. A 1500kg sedan is heading east and a 700kg subcompact is heading north. The subcompact has a green light and drives into the intersection. The tired driver of the sedan runs the red light and crashes directly into the driver’s side of the subcompact. The subcompact experiences a sudden, strong force to its right (toward the east).
a.Which car experiences the largest force due to the collision?
b.Which car experiences the largest acceleration due to the collision?
c.Which car is most likely to be thrown off the roadway because of the collision?
d.If the force on the subcompact is directly toward that car’s right (directly toward the east), what happens to the north/south component of the subcompact’s velocity?
11.A high-jumper and a long-jumper are both human falling objects, but they have slightly different goals. The high-jumper wants to travel over a bar without touching it and the long-jumper wants to travel as great a distance as possible without touching the ground.
a.The high jumper approaches the bar at a moderate forward velocity. She then jumps directly upward (she obtains a purely upward force from the ground). Why is her initial horizontal velocity important?
b.The long jumper approaches the jumping pit at peak sprint speed. He then jumps directly upward (he obtains a purely upward force from the ground). Why doesn’t he use all his strength to push himself forward instead of upward?
c.In a standing long-jump, the jumper leaps from a stationary position. Since the jumper has only so much strength and energy, she can only give herself a certain initial velocity. What direction should the jumper’s initial velocity have so that she travels the greatest distance?
d.The earth’s gravity is not perfectly uniform. Gravity is effectively 1% weaker at the equator than it is at the earth’s poles. Explain why both a high-jumper and a long-jumper benefit from weaker gravity.
12.Cable cars have traversed the hills of San Francisco for generations. As they ascend the hills, the cable cars are pulled up by moving, underground cables. As they descend the hills, the cable cars are held back by those same cables.
a.As the cable pulls the car uphill, which does (positive) work on which? As the cable lowers the car downhill, which does (positive) work on which?
*b.On a gentle hill, the car rises 1m in height as it travels 10m along the road. On a second, steeper hill, the car rises 1m in height as it travels 5m along the road. If the cable must exert 3,000N of force on the car to pull it up the gentle hill at constant velocity, how much force will be needed to pull the car up the steeper hill at constant velocity?
*c.The car now begins to descend the gentle hill. How much force is needed to keep the car from accelerating downhill and in which direction does the cable exert its force on the car?
*d.What is the weight of the cable car in newtons?
*e.What is the mass of the cable car in kilograms?
13.You’re piloting a very large, very massive oil tanker. The ship is so massive that it is barely affected by air and water resistance. If you turn off all the engines, it travels at nearly constant velocity through the water.
a.The ship is at rest in the water when you first turn the engines on full forward. The ship begins to push the water toward the east. Which way does the water push on the ship and which way does the ship accelerate?
*b.After 1minute of acceleration, the ship has reached a speed of 1km/h. If the force that the water exerts on the ship remains constant, how fast will the ship be traveling after another minute?
c.The ship has reached its cruising speed of 20km/h and you have steered it so that it is traveling directly northward. Its velocity is thus constant at 20km/h to the north. If you cause the ship to accelerate directly toward the west for a few minutes, in which direction will its new velocity point? (to the south? to the south-west? …?)
*d.A small sailboat cuts directly in front of your ship as you move forward at cruising speed. You immediately turn the engines on full reverse so that you now accelerate backward at the same rate you accelerated forward before. How long will it take your ship to stop?
e.Unfortunately, you collide with the sailboat. Each vessel exerts a force of 50,000N on the other vessel. Which vessel experiences the largest change in velocity and why?
14.Sledding makes use of the nearly frictionless nature of snow. Let’s look at how sledding works, assuming that the snow is truly frictionless and there is no air resistance.
*a.You’re pulling a child up a hill on a sled, traveling at a constant velocity. The child and sled weigh 400N and the smooth hill rises 1m for every 5m you travel along its surface. What is the net force on the child and sled?
*b.How much uphill force are you exerting on the child and sled?
*c.How much work must you do to pull the child and sled to the top of the 50m high hill?
*d.If the hill had been less steep but still 50m high, would you have had to do more work, less work, or the same amount of work to pull the child and sled to the top?
*e.How much has the child and sled’s gravitational potential energy increased in going from the bottom of the hill to the top?
*f.You release the child and sled and they slide down the hill, faster and faster. By the time they reach the bottom, how much kinetic energy do they have?
g.The sled continues on along the flat region at the bottom of the hill and onto some bare ground. The sled slows down abruptly and the child falls off the front. Why?
15.You are riding your bicycle on a north-south road and have just stopped at an intersection. Use that intersection as the reference point for position.
*a.You leave the intersection with a northward velocity of 3m/s. After 60s, how far are you from the intersection?
*b.What is your new position?
*c.You abruptly change your velocity to 3m/s toward the south. After another 60s, what is your position?
*d.Later in the day, you stop at the same intersection. This time you leave the intersection with an acceleration of 1m/s2 toward the north. After 10s, what is your velocity?
*e.What is your position?
*f.You abruptly change your acceleration to 1m/s2 toward the south. After another 10s, what is your velocity?
*g.What is your position?
h.A trip during which you reverse your velocity is evidently quite different from one in which you reverse your acceleration. Explain this difference.
16.Moving sidewalks are used in many airports to help people travel the long distances between gates. A moving sidewalk is essentially a conveyer belt for people. When you step on the sidewalk, its moving belt carries you along at a steady speed. Imagine a moving sidewalk that travels eastward at a speed of 1m/s. You step on it and quickly reach that same eastward velocity. You are now looking at the world from a new inertial frame of reference.