Chapter 2, third assignment

Answers to selected problems for Chapter 2 are found here.

Exercise 9. A rocket becomes progressively easier to accelerate as it travels through space. Why is this so? (Hint: about 90% of the mass of a newly launched rocket is fuel.)

As fuel is burned (and thrown overboard to provide thrust), the mass of the rocket decreases. This means that the same thrust produces an ever-increasing amount of acceleration, or, to say the same thing, less force is needed to keep a constant acceleration.
Exercise 19. Does a stick of dynamite contain force? Defend your answer.

Most physicists would say that a stick of dynamite does not contain force. It produces force when it is detonated. Until that time, it contains only potential force. (We will discuss potential energy in Chapter 3.)
Exercise 24. The strong man will push apart the two initially stationary freight cars of equal mass before he himself drops straight to the ground. Is it possible for him to give either of the cars a greater speed than the other? Why or why not?

No. Let us think about what happens when he tries to push harder on one car than the other. Remember that the two cars have the same mass.

Suppose he pushes on one car with a force of 800 N, and the other with a force of 1000 N (he's a strong man, after all!) The first car pushes back on him with a force of 800 N. Since the strong man is suspended between the two cars, the only thing he can brace on is the second car, which thus absorbs the reaction force of 800 N from the first car. Likewise, we can argue that the reaction force of 1000 N from the second car is absorbed by the first car. Ultimately, each car absorbs a total force of 1800 N. Since they have the same mass, they will experience the same acceleration and achieve the same speed at the moment when the strong man drops to the ground between them.
Exercise 35. How does the weight of a falling body compare with the air resistance it encounters just before it reaches terminal velocity? Just after?

Terminal velocity is the point at which air resistance (which is proportional to velocity) equals the weight of a falling object. Before the object reaches terminal velocity, the air friction is insufficient to completely stop acceleration and is therefore less than the object's weight. But once the object reaches terminal velocity, its speed no longer changes; air friction is equal to the object's weight from then on.
Exercise 37. If and when Galileo dropped two balls from the top of the Leaning Tower of Pisa, air resistance was not really negligible. Assuming that both balls were the same size yet one was much heavier than the other, which ball struck the ground first? Why?

The heavier ball would strike the ground first, because both balls would experience the same force of air resistance (proportional to size, not mass!) but the more massive one will not respond to that force as much because of its greater inertia.
  1. Both balls have the same downward acceleration due to gravity.
  2. Both balls experience the same upward force due to air friction, since that force is proportional to their (equal) cross-sectional area. But...
  3. The upward acceleration due to air friction, which subtracts from the gravitational acceleration, is inversely proportional to the mass of each ball: a = F/m. Therefore...
  4. The more massive ball experiences a smaller upward frictional acceleration, and so the net downward acceleration (gravity minus friction) is larger for the more massive ball.
  5. Therefore, the more massive ball strikes first.
Exercise 39. Which is more likely to break, the ropes supporting a hammock stretched tightly between a pair of trees or one that sags more when you sit on it? Defend your answer.

A tightly-stretched hammock will break no sooner than a saggy one. Here's why:

A rope breaks when it reaches its breaking strain. If the breaking strain is 800 N and you weigh 900 N, the rope will break after stretching as far as it can stretch. If the rope is already fully stretched, it will break without sagging... but the loose rope will still break after it is fully stretched, unless there's enough sag in it for you to touch the ground before the rope is tightened up by your weight.