Physics 30S
Gravitational Fields
Problem Package
Terminal Velocity
The terminal velocity of a Ping-Pong ball in air is only 9 m/s. A basketball has a terminal velocity of 20 m/s, while a baseball can fall as fast as 42 m/s. Skiers increase their terminal velocities by decreasing drag force. They hold their bodies in an "egg" shape and wear very smooth clothing and streamlined helmets. A skydiver can control terminal velocity by changing body shape. A spread-eagle position gives the slowest terminal velocity, about 60 m/s. By opening the parachute, the skydiver has become part of a very large object with a correspondingly large drag force. The terminal velocity is now about 5 m/s.
Concept Review Questions
1. A sky diver in the spread-eagle position opens the parachute. Is the diver accelerated? Which direction? Explain your answer.
2. A weight lifter lifts a 115-kg barbell from the ground.
a.) How does the force exerted by the lifter compare with the weight of the barbell? Explain.
b.) How does the force exerted by the lifter on the barbell compare with the force exerted by the barbell on the lifter?
3. Suppose you put a wooden block against a wall and push horizontally to keep it from falling. Draw a diagram showing the forces on the block and the causes of those forces. Pay special attention to the normal force and the weight of the block. Are they in the same direction? Are they equal?
4. Explain why an aluminum ball and a steel ball of similar size and shape, dropped from the same height, reach the ground at the same time.
5. Give some examples of falling objects for which air resistance cannot be ignored.
6. Give some examples of falling objects for which air resistance can be ignored.
7. The value of “g” on the moon is one-sixth of its value on Earth.
a. Will a ball dropped from the same height by an astronaut hit the ground with a smaller or larger speed than on Earth?
b. Will the ball take more or less time to fall?
8. A ball is thrown vertically upward with the same initial velocity on Earth and on planet Alpha, which has three times the gravitational acceleration as Earth.
a. How does the maximum height reached by the ball on Alpha compare with the maximum height on Earth?
b. If the ball on Alpha were thrown with three times greater initial velocity, how would that change your answer to a?
9. Rock A is dropped from a cliff; rock B is thrown downward.
a. When they reach the bottom, which rock has a greater velocity?
b. Which has a greater acceleration?
c. Which arrives first?
Problems
1. Suppose Joe, who weighs 625N, stands on a bathroom scale calibrated in Newtons.
a. What force would the scale exert on Joe? in what direction?
b. If Joe now holds a 50-N cat in his arms, what force would the scale exert on him?
c. After Joe puts down the cat, his father comes up behind him and lifts upward on his elbows with a 72-N force. What force does the scale now exert on Joe?
2. If you drop a golf ball, how far does it fall in 0.5 s? (-1.2 m)
3. A spacecraft traveling at a velocity of +1210 m/s is uniformly accelerated at - 150 m/s2. If the acceleration lasts for 8.68 s what is the final velocity of the craft? Explain your results in words. (-90 m/s)
4. Table 4-1 is a table of the displacements and velocities of a ball at the end of each second for the first 5.0 s of free-fall from rest.
a. Use the data in the table to plot a velocity-time graph.
b. Use the data in the table to plot a position-time graph.
c. Find the slope of the curve at the end of 2.0 and 4.0s on the position-time graph. What are the approximate slopes? Do the values agree with the table of velocity?
d. Use the data in the table to plot a position versus time-squared graph. What type of curve is obtained?
e. Find the slope of the line at any point. Explain the significance of the value.
f. Does this curve agree with the equation?
d = ½g2
Table 4-1
time (s) / displacement (m) / velocity (m/s)0 / 0 / 0
1 / -4.9 / -9.8
2 / -19.6 / -19.6
3 / -44.1 / -29.4
4 / -78.4 / -39.2
5 / -122.5 / -49
5. A stone falls freely from rest for 8.0 s.
a. Calculate the stone's velocity after 8.0 s.
b. What is the stone's displacement during this time?
6. An object shot straight up rises for 7.0 s before it reaches its maximum height. A second object falling from rest takes 7.0 s to reach the ground. Compare the displacements traveled by the objects during the 7.0-s period.
7. Kyle is flying a helicopter when he drops a bag. When the bag has fallen 2.0 s,
a. What is the bag's velocity?
b. How far has the bag fallen?
8. Kyle is flying the same helicopter and it is rising at 5.0 m/s when he releases the bag. After 2.0 s,
a. What is the bag's velocity?
b. How far has the bag fallen?
c. How far below the helicopter is the bag?
9. Kyle's helicopter now descends at 5.0 m/s as he releases the bag. After 2.0 s,
a. What is the bag's velocity?
b. How far has the bag fallen?
c. How far below the helicopter is the bag?
d. What is common to the answers to Problems 7, 8 & 9?
10. A weather balloon is floating at a constant height above Earth when it releases a pack of instruments.
If the pack hits the ground with a velocity of -73.5 m/s, how far does the pack fall?
11. A 95.0-kg boxer has his first match in the Panama Canal Zone (g = 9.782 m/s2) and his second match at the North Pole
(g = 9.832 m/s2 ).
a. What is his mass in the Canal Zone?
b. What is his weight in the Canal Zone?
c. What is his mass at the North Pole?
d. What is his weight at the North Pole?
e. Does he "weigh-in" or does he really “mass-in"?
12. A rubber ball weighs 49 N.
a. What is the mass of the ball?
b. What is the acceleration of the ball if an upward force of 69 N is applied?
13. A small weather rocket weighs 14.7 N.
a. What is its mass?
b. A balloon carries the rocket up. The rocket is released from the balloon and fired, but its engine exerts an upward force of 10.2 N. What is the acceleration of the rocket?
14. The space shuttle has a mass of 2.0 x 106-kg. At lift-off the engines generate an upward force of 30 x 106 N
a. What is the weight of the shuttle?
b. What is the acceleration of the shuttle when launched?
c. The average acceleration of the shuttle during its 10 min launch is 13 m/s2. What velocity does it attain?
d. As the space shuttle engines burn, the mass of the fuel becomes less and less. Assuming the force exerted by the engines remains the same, would you expect the acceleration to increase, decrease, or remain the same? Why?
15. A 4500-kg helicopter accelerates upward at 2 m/s2. What lift force is exerted by the air on the propellers?
16. The maximum force a grocery bag can withstand and not rip is 250 N. If 20 kg of groceries are lifted from the floor to the table with an acceleration of 5 m/s2, will the bag hold?
17. A student stands on a bathroom scale in an elevator at rest on the 64th floor of a building. The scale reads 836 N.
a. As the elevator moves up, the scale reading increases to 935 N, then decreases back to 836 N. Find the acceleration of the elevator.
b. As the elevator approaches the 74th floor, the scale reading drops as low as 782 N. What is the acceleration of the elevator?
c. Using your results from parts a and b, explain which change in velocity, starting or stopping, would take the longer time.
d. Explain the changes in the scale you would expect on the ride back down.
18. The instruments attached to a weather balloon a mass of 5.0 kg.
a. The balloon is released and exerts an upward force of 98 N on the instruments. What is the acceleration of the balloon and instruments?
b. After the balloon has accelerated for 10 s, the instruments are released. What is the velocity of the instruments at the moment of their release?
c. What net force acts on the instruments after their release?
d. When does the direction of their velocity first become downward?
Answers
1)
a) 625 N up
b) 675 N up
c) 553 N up
2) –1.23 m
3) –92 m/s
4) on graph paper
5)
a) –78.4 m/s
b) 313.6 m
6) 240.1 m exactly the same
7)
a) –19.6 m/s
b) –19.6 m
8)
a) –14.6 m/s
b) –9.6 m
c) 19.6 m
9)
a) –24.6 m/s
b) –29.6 m
c) 19.6 m
10) 275.6 m
11) a) 95 kg b) 929.29 N c) 95 kg d) 934.04 N e) mass
12) 5 kg b) 4 m/s2
13) 1.5 kg b) –3 m/s2
14) 1.96 x 107 N b) 5.2 m/s2 c) 7800 m/s d) a is more
15) 53100 N
16) 296 N NO
17) a) 1.16 m/s2 b) -.63 m/s2 c) stopping d) weigh less, then more
18) a) 9.8 m/s2 up b) 98 m/s c) only gravity d) 10 seconds later