Projectile Compresses a Spring Gun

[BrandonInFlorida]

Homework Statement

A ball of mass m which is projected with speed vi into the barrel of a spring-gun of mass M initially at rest on a frictionless surface, as shown in the attached file below. The ball sticks in the barrel at the point of maximum

compression of the spring. No energy is lost in friction.

A) In terms of the given masses and the kinetic energy, what energy is stored in the spring at its maximum compression?

B) If the mass of the ball and the gun are equal and the spring constant is given as k, determine the maximum compression of the spring in terms of the initial kinetic energy and the spring constant k.

Relevant Equations

m1v1i = m1v1f + m2v2
Kinetic Energy=(1/2)mv2
Spring Energy=(1/2)kx2

This is a common homework problem and I did find a post here that talks about it, but that post was closed to comments, so I am reproducing it to be able to ask a question.

We are, apparently, according to solutions I have found, supposed to recognize that it is an inelastic collision, since the ball sticks to its target, use conservation of momentum to find the velocity of the final bullet + gun system, use that velocity to calculate the kinetic energies before and after the collision, and then assume the difference goes to compress the spring.

Here is my question. We are told that in inelastic collisions, mechanical energy is not conserved with some of the energy going to heat. Yet now we are told that all of the energy which is no longer kinetic, goes into the spring. How am I supposed to know that for the very first time in any inelastic collision problem I have ever seen, no energy goes to heat and mechanical energy is conserved, and, for that matter, why is it true here, since it has never been true in any other inelastic collision problem of the many I have worked?

 

[rashida564]

In elastic collisions, KE energy is conserved. And in inelastic collision KE is not conserved it may go into heat, or may go completely to PE, but at the end KE is not the same before and after the collision

 

[BrandonInFlorida]

Thank you. I appreciate your response, but since KE has gone partially to heat in every inelastic collision I've ever seen in any physics text, how was I (or anyone), seeing this problem, supposed to conclude that here no energy at all goes to heat? And, actually, why does this system behave differently from every other inelastic collision I've seen and send no energy to heat? I am not sure how the student is supposed to conclude that he need not worry about the initial projectile's energy going partially to heat. Until one knows that that's off the table, one cannot solve this.

 

[rashida564]

It's called the element of surprise, the goal of it is to make students lose marks

 

[gneill]

From the original problem statement:

The ball sticks in the barrel at the point of maximum
compression of the spring. No energy is lost in friction.

[My highlighting]

 

[BrandonInFlorida]

Gneill, it isn't a question of friction. That statement is intended to let you know that there are no additional frictional losses, e.g. in the spring motion or the ball scraping the wall or something. In an inelastic collision, such as one in which a piece of gum is fired at an object and sticks to it, energy is lost to heat and I have never once heard anyone call that friction. In every completely inelastic collision I've ever heard of, substantial energy is lost to heat. How are we supposed to magically know that that doesn't happen here and that now the missing kinetic energy is going 100% into compressing the spring?

 

[gneill]

Presumably the spring in the "spring-gun" is ideal. As there is no mention to the contrary you can assume that it is so. If the ball did not stick at the point of maximum compression, then the spring would have launched the ball back out and the collision would have been perfectly elastic. So in this case where the ball sticks at maximum compression, you've essentially got one half of a perfectly elastic collision.

 

[BrandonInFlorida]

I can see that you know a lot, but the idea that if the ball didn't stick, the collision would have been elastic, seems like something you could say that about any inelastic collision. The ball did stick. If this is, as you say, an elastic collision, I don't think it's very obvious. This problem seems to be more complicated than the discussions I have seen today, checking this on the Web, admit.

 

[TSny]

The ball did stick. If this is, as you say, an elastic collision, I don't think it's very obvious.

Yes, I agree. It's not all that obvious when seeing it for the first time. I don't know if it would help to consider in more detail the collision of the ball with the free end of the spring as shown:

Let the free end of the spring have a little plate (shown in brown) that the ball can stick to. Suppose for now that this plate has a mass $m'$. You can consider the collision to be a completely inelastic collision between the ball of mass $m$ and the plate. The spring has no influence on this collision between the ball and the plate since we can consider the collision as instantaneous. The spring does not compress any during the collision, so it does not exert any force during the collision.

Thus, the collision can be thought of as a completely inelastic collision between the ball and a freely moving mass $m'$ initially at rest. Using conservation of momentum, you can work out an expression for the speed of the ball-plate system immediately after the collision. You will find, as expected, that KE is not conserved in the collision. Some of the KE has been converted into "heat" or internal energy of the ball and plate.

However, if you let the mass of the plate $m'$ approach zero, you will see from the equations that the final speed of the ball-plate system just after the collision is the same as the initial speed of the ball. And you will see that no KE is lost in the collision of the ball with the end of the spring. So, no "heat" is generated when the ball makes contact with the spring even though the ball "sticks" to the end of the spring. So, mechanical energy is going to be conserved overall as the ball hits the spring and then compresses the spring.

 

[gneill]

@TSny: Ugh. I kind of hate it when others express more eloquently the thoughts I had but didn't take the time to fully express myself. I must be getting lazy in my old age. That's definitely on me. Apologies to the OP for not being thorough in my own replies.

 

[BrandonInFlorida]

This is a good discussion. So, the trick is to view this as a collision between the ball and a massless object that won't slow it down, and to regard the subsequent compression of the spring as an unconnected later event. I hope that someone has tested this experimentally and verified that this is a correct interpretation.

I got this out of "Halliday and Resnick volume 1" (circa 1962). Other people give exactly or essentially the same problem, but I think it's stolen from H & R. H & R present this as being a medium level of difficulty, but I think there are two ways a student could get this right. The first way would be to be oblivious to the whole issue of heat loss in an inelastic collision and just reflexively apply conservation of mechanical energy. The second way would be to use the logic you have suggested, which I think is a lot to ask of a student in an introductory class.

Thanks so much, everybody!

Brandon