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I'm working on a physics problem in preparation for the MCAT and there's this particular problem that's troubling me. I don't know if it's a bad question or if I'm not understanding some sort of concept. I was hoping someone here can clarify. Here's the problem verbatim:

A 1kg cart travels down an inclined plane at 5 m/s and strikes two billiard balls, which start moving in opposite directions perpendicular to the initial direction of the cart. Ball A has a mass of 2kg and moves away at 2m/s and ball B has a mass of 1kg and moves away at 4m/s. Which of the following statements is true?

a) the cart will come to a halt in order to conserve momentum

b) the cart will slow down

c) the cart will continue moving as before, while balls A and B will convert the gravitational potential energy of the cart into their own kinetic energies

d) these conditions are impossible because they violate either the law of conservation of momentum or conservation of energy

At first glance, it appears to me that the answer is (D) because the system seemingly has more total momentum after the collision than before the collision. However, the answer explanation insists the correct answer to be (C) since it claims that "kinetic energy is not conserved; the system gains energy in this inelastic collision".

I can understand that this gain in energy can come from gravitational potential energy from the incline the cart is on; however, it is ambiguous if the cart is accelerating down the incline. In order for the scenario to be consistent with choice (C), does the cart have to be accelerating down the incline? Or do you take the problem to mean that the cart is leaving the incline at 5m/s? Or am I missing or not understanding something?

How would you interpret this problem and which explanation do you think is most consistent with the scenario? What assumptions do you have to make to arrive at your answer?

The answer key's explanation, verbatim is as follows:

The law of conservation of momentum states that both the vertical and horizontal components of momentum for a system must stay constant. If you take the initial movement of the cart as horizontal and the two balls move in perpendicular directions to the horizontal, it means that the cart must maintain its horizontal component of velocity. Therefore, (A) and (B) are wrong. If the billiard balls move as described, then kinetic energy is not conserved; the system gains energy in this inelastic collision. (C) correctly describes how this scenario is possible.

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Hi jpan, and welcome to Physics Stack Exchange! Good question :-) I'm adding the "homework" tag because that tag applies to all problems of an educational nature, not just actual homework assignments. –  David Z Jan 8 '12 at 2:38
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I think the problem as worded is a bit vague. Since you are told the balls travel in a direction perpendicular to the motion of the cart, momentum conservation rules out a and b. I need to look at this in the morning when I'm not tired. –  WWright Jan 8 '12 at 3:01
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3 Answers

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It's a bad question. For one thing, answer (C) is utter nonsense. (Maybe that's a bit harsh. It might be just regular nonsense.) In order for something to convert gravitational potential energy into kinetic energy, it has to drop to a lower height under the influence of gravity. This does not happen during a collision. Collisions in physics are effectively instantaneous events; they occur at one point in space and time and then they're over and done with. There is no change in height by which GPE could be converted into KE during the collision. Whatever (kinetic) energy the balls run away with, they had to obtain it from the kinetic energy that the cart had coming into the collision.

Now, the kinetic energy of the cart at the point of the collision was converted from the gravitational potential energy that the cart had higher up the ramp. But that conversion was done by the cart alone; the balls had nothing to do with it.

The other reason I don't like this problem is that they don't tell you at which point on the ramp the cart has the speed of $5\text{ m/s}$. It's possible that the cart maintains a constant velocity as it goes down the incline, but that would require some mechanism to keep the cart from accelerating, and if some such mechanism is involved, it should be mentioned in the problem. If that is the case, the gravitational potential energy that the cart started out with would have been converted into some other form of energy, not kinetic. It might be heat, electricity, spring energy, etc. but there's no way to know unless they tell you what mechanism is keeping the cart from accelerating.

In a pinch, if you encountered this problem on the test and didn't have any opportunity to ask for clarification, I would just assume that $5\text{ m/s}$ is the speed at the end of the ramp, immediately prior to when the cart hits the balls. Why? The alternative is that the problem is unsolvable. If the speed of the cart coming into the collision is not $5\text{ m/s}$, you have no other information that would allow you to calculate what it is. (Self-check: do you understand why this is the case?)

Once you assume that the speed of the cart coming into the collision is $5\text{ m/s}$, you have a collision of 3 objects, each of which has a mass and initial and final velocities. All 3 masses, all 3 initial velocities, and two of the final velocities are known, so you should have enough information to solve for the third. If you don't find any solution, then the situation is impossible and the answer is (D); on the other hand, if you do find a solution for the final velocity of the cart, then that velocity will distinguish between choices (A) ($v_f = 0$), (B) ($v_f < 5\text{ m/s}$), and (C) ($v_f = 5\text{ m/s}$, if you ignore the stuff about energy being converted).

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Thanks! That's a very through explanation. Another thing to mention: I discussed this problem with a friend and he said that the problem's scenario should be impossible. Assuming the two balls are initially at rest, no matter what shape the front of the cart, it is impossible for the cart to hit the balls and have the balls recoil exactly perpendicularly to the impact. the only way this would be possible is if the balls had some initial velocity, with a component anti-parallel to the direction of the cart, that got exactly cancelled by part of the cart's momentum along that direction. –  jpan Jan 8 '12 at 4:29
    
(continued) This results in the balls resulting net momentum being exactly perpendicular to the cart's (now reduced) momentum along its original direction. I was wondering what your thoughts are on this. –  jpan Jan 8 '12 at 4:30
    
Yeah, I had the same thought. I didn't mention it because I figured that among the various things that are wrong with this problem, that's one of the less important ones, but your friend's argument seems perfectly legit. –  David Z Jan 8 '12 at 4:44
    
The correct answer is D. It's not clear to me whether C is an officially sanctioned answer, or whether the author of the book this question was in just collected the questions and figured out the answers by himself (in this case, incorrectly). I suspect the second. –  Peter Shor Jan 8 '12 at 15:08
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I am sorry but you all got it wrong. In short, the answer is B.

There are 4 mobile objects in the system. 1. The cart, 2. Ball A, 3. Ball B, and 4. The Earth (Yes! The whole Earth itself).

Say the vertical axis is z, horizontal axis along which the balls moved is x, and the other horizontal axis perpendicular to x is y. The cart is moving in y-z plane. The momentum along axes y and z is conserved by the motion of the Earth in the direction opposite to cart. Unbelievable that the Earth moves in response to objects moving in its vicinity isn’t it? But the fact is it does move. Since the Earth is massive, its velocity in this regard is extremely small, imperceptible and negligible.

Momentum along x-axis is zero and is conserved as you can easily calculate.

Now coming to the energy part, clearly the balls gained energy and it has to come from somewhere. Well, it comes from the kinetic energy of the cart-Earth system. They both slow down after the impact and impart the kinetic energy they lost to the balls.

Now, how does the Earth know when the cart slowed down and adjust its velocity accordingly? That’s the beauty of gravity. It is invisible and yet makes its presence felt.

Take it as an exercise and calculate the velocities of the Earth before and after the impact and the velocity of cart after impact. It will be fun.

Here are the answers (Assumption: relative velocities of Earth and cart $= 0$ before the cart is released down the incline and the impact is elastic)

Velocity of Earth before impact = $8.372 \cdot 10^{-25}$ m/s

Velocity of Earth after impact = $1.674 \cdot 10^{-25}$ m/s

Velocity of cart after impact = $1.000$ m/s

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If you consider ball A and ball B, they each have a momentum of 4 kg m/s , but in opposite directions, for a total momentum of 0. This means that the velocity of the cart cannot be changed by the collision, from the law of conservation of momentum. This in turn means that the kinetic energy is increased by the collision, violating the law of conservation of energy.

The correct answer should be D.

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