Chapter 3, first assignment

Answers to selected problems for Chapter 3 are found here.

Exercise 2. To bring a supertanker to a stop, its engines are typically cut off about 25 km from port. Why is it so difficult to stop or turn a supertanker?
A supertanker is very, very massive; that means that its momentum -- at any speed! -- will be very large. A large momentum requires a large impulse to change or cancel it.
Exercise 4. Why might a wine glass survive a fall onto a carpeted floor but not onto a concrete floor?
The falling wine glass requires a certain amount of impulse to stop it. But it takes a certain amount of force to break the glass. Impulse is force × time; therefore, if we have a surface (like carpet) that can give a bit, an impact will take longer and produce less force. Concrete, on the other hand, has almost no give at all, so any impact will have a much larger force than the same impact with carpet.
Exercise 7. Why is a punch more forceful with a bare fist than with a boxing glove?
The boxing glove has much more give than a fist, spreading out the impulse of the punch over more time and thus producing a lower impact force. (You can break your knuckles hitting someone in the jaw with your bare fist, but punches to the jaw are commonplace in boxing.)
Exercise 8. A boxer can punch a heavy bag for more than an hour without tiring, but will tire quickly when boxing with an opponent for a few minutes. Why? (Hint: When aimed at the bag, what supplies the impulse to stop the punches? When aimed at the opponent, what or who supplies the impulse to stop the punches that are missed?)
The momentum of the heavy bag is very large, and so it can absorb a lot of impulse without moving much. If the boxer connects with his opponent, the same holds true (though the opponent is probably lighter than the boxing bag!) But if the boxer misses, he has to stop his fist with his own muscle power. That's why shadowboxing is such a good workout!
Exercise 12. Why is it difficult for a firefighter to hold a hose that ejects large quantities of water at a high speed?
Action and reaction. The water, which is gaining a very large momentum as it is ejected from the hose, imparts a reaction impulse to the hose itself. Since the ejection time is very short, the reaction force is very large.
Exercise 16. Would a head-on collision between two cars be more damaging to the occupants if the cars stuck together or if the cars rebounded upon impact?
Bounces provide a greater impulse in collisions than simply sticking to the struck object. Therefore, a rebounding collision imparts a greater impulse (and is more damaging) than a collision in which the two cars stick together.
Problem 2. A railroad diesel engine weighs four times as much as a freight car. If the diesel engine coasts at 5 km per hour into a freight car that is initially at rest, how fast do the two coast after they couple together?
The freight car has mass m, the diesel engine has mass 4m. Conservation of momentum tells us that the total momentum after the coupling is the same as before. But momentum equals mass × velocity. Therefore (since the initial momentum of the freight car is zero and the total mass after coupling is 5m)

4mvinitial = 5mvfinal

4mvinitial  = vfinal

5m

4  vinitial = vfinal

5

and the final velocity is (4/5)×5 km/h = 4 km/h.
Problem 3. A 5-kg fish swimming at 1 m/s swallows an absent-minded 1-kg fish at rest.
  1. What is the speed of the larger fish after lunch?
    This can be answered in the same way as Problem 2. The total mass after lunch is 6 kg, and the momentum (before and after) is 5 kg × 1 m/s, so the speed of the larger fish after lunch is 5/6 of a meter per second.
  2. What would be its speed if the smaller fish were swimming toward it at 4 m/s?
    Now the total momentum of the two fish is (5 kg × 1 m/s) minus (1 kg × 4 m/s). 5 minus 4 equals 1 kg m/s, so the velocity of the big fish after lunch will now be 1/6 of a meter per second.