Why is this inelastic collision problem solvable?

[Nathanael]

This problem comes from the inelastic collision section.

Homework Statement

"In Fig. 9-63, block 1 (mass 2.0 kg) is moving rightward at 10 m/s and block 2 (mass 5.0 kg) is moving rightward at 3.0 m/s. The surface is frictionless, and a spring with a spring constant of 1120 N/m is fixed to block 2. When the blocks collide, the compression of the spring is maximum at the instant the blocks have the same velocity. Find the maximum compression."

"Fig. 9-63" is just two blocks moving to the right. The block on the right side ("block 2") has a spring attatched to the left (or "back") of it.

The Attempt at a Solution

I realize you can solve it by solving for the kinetic energy lost when they're moving at the same speed (the problem kind of ruined the fun with the hint) but my question isn't about.

My question is, why is safe to assume that no energy is lost by other means? (via thermal energy, sound waves, deformation, or whatever else)

The problem never said anything about the collision being "ultimately elastic" (and it is in the section "inelastic collisions")

Wouldn't you need to know that the collision is "ultimately elastic" (meaning kinetic energy is conserved in the end)?



Are they just assuming energy loss is negligable? (That would kind of defeat the purpose of the collision being "inelastic")

Or is there something inherent about colliding into a spring that implies that the collision is (at least approximately) "ultimately elsastic"
(that somewhat makes sense, since the spring would "absorb some of the impact" so-to-speak, but I don't understand why they would expect you to make that assumption)

Am I missing something here?

 

[KurtWagner]

Perhaps it is inelastic in the way that the spring absorbs energy.

The problem seems to finish before the spring gives the energy back so you need to work it out as though some of the kinetic energy is lost to the potential elastic energy of the spring...

I think

 

[Nathanael]

Yeah, I was thinking that. But in a way, that makes this not even a collision problem (it's only half of a collision).

I guess they just wanted you to assume energy lost is negligable, and I'm just getting caught up on that because it's an uncommon assumption for inelastic collisions. I suppose though, in this situation, it's not too bizarre.

Is it generally the case that (for a fixed change in momentum) the longer the impulse acts, the less energy is "lost" to various forms?

I'm guessing that is the case.
(Because "softer" materials that sort of "give in" (causing a longer impulse without increasing total ΔP) cause less deformation and typically less noise (and probably less heat) than a "rigid" material)

A spring is essentially an exaggerated form of a material that "gives in." This sort of explains why this problem is a unique case of an inelastic collision in which it is probably safe to assume negligable energy loss.
(Although it still is odd to me)

 

[KurtWagner]

Do you have the answer from the book (or whateva)?

 

[Nathanael]

Sure, it's a quarter of a meter.

 

[KurtWagner]

I got 0.30m, so I must be doing something wrong :p

 

[Nathanael]

The reasoning goes loosely like this:

The maximum compression is when they're going at the same speed (the reason is that if there was a relative velocity between the two masses, the spring would either be "getting more compressed" or "getting less compressed")

Since momentum is conserved, you can find what speed they will be going when they are traveling at the same speed.

You can then calculate the kinetic energy at that time, and compare it to the original kinetic energy, and the difference should be the spring's potential energy.

 

[AlephZero]

My question is, why is safe to assume that no energy is lost by other means? (via thermal energy, sound waves, deformation, or whatever else)

Don't confuse the simple models of how things behave in textbook problems with the real world. The ideas of perfectly elastic and perfectly inelastic collisions are simple assumptions that make problems easy to solve. They don't necessarily correspond to a particular real-word situation. Of course they are good approximations to many real world situations.

Wouldn't you need to know that the collision is "ultimately elastic" (meaning kinetic energy is conserved in the end)?

The question only refers to the first part of the collision, when the structure is absorbing energy by compressing the spring. What happens to the energy in the spring when (or even if) it is released is another question.

In "real life" all collisions take a finite amount of time, like the one in this question. But if you assume the bodies that collide are perfectly rigid, you can ignore that fact and assume the changes in momentum and/or energy happen instantaneously.

 

[Chestermiller]

The problem statement inherently assumes that the spring is ideally elastic when they say that its spring constant is 1120 N/m. If its behavior were more complicated than that, they would have to provide a more complicated relationship. The two masses don't actually make direct contact, so there is no in-elasticity involved with that.

Chet

 

[Nathanael]

Thankts Chet, that was helpful.