Show mechanical energy is not conserved in an inelastic collision.

[brikayyy]

Homework Statement

Using the equation for the final velocity of an inelastic collision, show that mechanical energy is not conserved in the collision. (Do this only with variables.)

Homework Equations

V_f = (m₁v_1i + m₂v_2i)/(m₁ + m₂)
KE = (1/2)mv²
TME = KE + PE

The Attempt at a Solution

I honestly have no idea where to begin. I thought that maybe I could somehow use the equation of an elastic collision, but the problem says to use the one for inelastic collision.

Should I begin by rewriting the equation to look something like this?

V_f = K_f/K_i = [(1/2)m₁v_1i + (1/2)m₂v_2i]/[(1/2)(m₁ + m₂)]

TME has to equal zero if it's not conserved, right?

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Thanks for any help provided!

 

[tia89]

I assume that the case you are working on is the one in which after the collision the two bodies stick together right?? This is the classic case for inelastic collisions... and is compatible with the formula you give for the final velocity...

At this point you can just compute the final and initial kinetic energy... and for the final one you have an expression of the final velocity as function of the initial ones... you will easily see doing the math that this is different from the initial kinetic energy

 

[brikayyy]

I assume that the case you are working on is the one in which after the collision the two bodies stick together right?? This is the classic case for inelastic collisions... and is compatible with the formula you give for the final velocity...

At this point you can just compute the final and initial kinetic energy... and for the final one you have an expression of the final velocity as function of the initial ones... you will easily see doing the math that this is different from the initial kinetic energy

Thanks for replying! Yes, it's completely safe to assume that. :)

I will try that, but I'm kind of confused as to why I'm computing kinetic energy? I thought that total mechanical energy would have to equal zero if it wasn't conserved, theoretically speaking.

 

[rcgldr]

I will try that, but I'm kind of confused as to why I'm computing kinetic energy? I thought that total mechanical energy would have to equal zero if it wasn't conserved, theoretically speaking.

Zero total mechanical energy after an inellastic collision only occurs if after the collision neither object is moving. In the more common case, total mechanical energy is reduced, but not to zero.

 

[CAF123]

Thanks for replying! Yes, it's completely safe to assume that. :)

I will try that, but I'm kind of confused as to why I'm computing kinetic energy? I thought that total mechanical energy would have to equal zero if it wasn't conserved, theoretically speaking.

The final equation for v suggests a collision on a level straight, so TME = KE. If TME is not conserved, then TME of the two body system before the collision is not equal to TME afterwards.

For a conserved system, TME = const. so the total does not change with time.

 

[tia89]

I thought that total mechanical energy would have to equal zero if it wasn't conserved, theoretically speaking.

If something is indeed moving his mechanical energy will always be different from zero, as it will have some kinetic energy. Also in your case the ONLY way to have $v_f=0$ is that the initial velocity of one of the two bodies is already zero AND the mass of that object is infinite, OR the collision is head on and they have same mass and same velocity.

Now in your problem you have no indication of any potential energy involved, so mechanical energy reduces to kinetic energy. Non-conservation of that doesn't mean that final kinetic energy is zero (they would have to stop, so t is not general case) but that it is different from the initial kinetic energy. Do the math as I suggested and compare with the initial kinetic energy (simply the sum of the two initial kinetic energies) and you will see that this is the case.

 

[CWatters]

One proof I've seen sets the starting velocity of one mass (say mass 2) to zero. It justifies this approach by saying if it wasn't zero you could make it so by using a co-ordinate system that moves at the initial velocity of mass 2.

So start with one mass stationary (eg V_2i = 0) then..

Write an equation for conservation of momentum.
Write equations for the before and after KE.
Express as a ratio (eg KE Before/KE After)
Some things cancel
You are left with a ratio that must be >1.

 

[rcgldr]

If something is indeed moving his mechanical energy will always be different from zero, as it will have some kinetic energy. Also in your case the ONLY way to have $v_f=0$ is that the initial velocity of one of the two bodies is already zero AND the mass of that object is infinite, OR the collision is head on and they have same mass and same velocity.

Or on a more general case, a head on collision where the magnitude of momentum of each object is the same: m1 v1 = - m2 v2,