Completely Inelastic Collision

[MathewsMD]

I'm currently trying to make a proof to convince myself that when two object collide and stick afterwards, there is maximum energy loss. I've been thinking about it and trying to come up with a mathematical proof to solidify the idea in my head.
Please tell me if there's any errors in my explanation or if there's anything that should be added.

Case 1: object 1 is moving and object 2 is stationary (with no external forces, a frame of reference can always be used in which the motion is 0 m/s, and I realize this is a proof in itself, but I want to come up with something mathematically instead of intuitively)

K_i = (1/2)m₁v_i² [1]

Taking the derivative and solving for 0 will give me an extreme value for the kinetic energy.

K_f = (1/2)(m₁+m₂)v_f² [2]

K'_f=p=(m₁+m₂)v_f
and if v_f=0 m/s, then this system will have 0 J (which, using the right frame of reference, is possible in any situation where the velocity of the two "stuck" objects is constant"

p_i = p_f since there is no net external force
m₁v_i=(m₁+m₂)v_f
v_f=m₁v_i/(m₁+m₂) [3]

Plugging [3] into [2] and dividing by [1], to see the ratio between K_f and K_i

=[(1/2)(m₁)[m₁²v_i²/(m₁+m₂)²]/(1/2)m₁v_i²

=m₁²/(m₁+m₂)²

I'm just confused now since this ratio doesn't seem to tell me much about two kinetic energies, does it? What else should I do now?

 

[Nugatory]

I'm just confused now since this ratio doesn't seem to tell me much about two kinetic energies, does it? What else should I do now?

Try working the problem in a frame in which the total momentum is zero. Then, before you go through the work of finding the extreme value, look carefully at the pre-collision and post-collision state.

 

[mic*]

Firstly, where K is kinetic energy, m is mass, v is velocity, and p is momentum, it could be technically debated that, if:
K_f = (1/2)(m₁+m₂)v_f² [2]
then:
K'_f ≠ p = (m₁+m₂)v_f
What has K been differentiated with respect to?

Another more easily addressed problem appears to be;
p_i = p_f
This is true if momentum is conserved, as is the case with ideal elastic collisions. In an inelastic collision, KE from equation [1] in the OP is lost to heat and other processes involved in the coalescing of the two objects, regardless of whether the final velocity; has been fixed to be zero, is measured to be zero, or otherwise. So p_i ≠ p_f , and therefore equation [3] is incorrect.

Lastly if K is dependent on v, and the final velocity of the coalesced objects is 0m.s^-1, it becomes clear that the ratio K_f / K_i provides no useful information.

 

[Nugatory]

This is true if momentum is conserved, as is the case with ideal elastic collisions.

Momentum is always conserved, in both elastic and inelastic collisions, whether ideal or not.

 

[dauto]

Use the center of mass reference frame. In this frame the objects are at rest after the collision (by momentum conservation) which means the energy is a minimum (since its zero, and cannot be negative).