- #1
MathewsMD
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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)
Ki = (1/2)m1vi2 [1]
Taking the derivative and solving for 0 will give me an extreme value for the kinetic energy.
Kf = (1/2)(m1+m2)vf2 [2]
K'f=p=(m1+m2)vf
and if vf=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"
pi = pf since there is no net external force
m1vi=(m1+m2)vf
vf=m1vi/(m1+m2) [3]
Plugging [3] into [2] and dividing by [1], to see the ratio between Kf and Ki
=[(1/2)(m1)[m12vi2/(m1+m2)2]/(1/2)m1vi2
=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?
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)
Ki = (1/2)m1vi2 [1]
Taking the derivative and solving for 0 will give me an extreme value for the kinetic energy.
Kf = (1/2)(m1+m2)vf2 [2]
K'f=p=(m1+m2)vf
and if vf=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"
pi = pf since there is no net external force
m1vi=(m1+m2)vf
vf=m1vi/(m1+m2) [3]
Plugging [3] into [2] and dividing by [1], to see the ratio between Kf and Ki
=[(1/2)(m1)[m12vi2/(m1+m2)2]/(1/2)m1vi2
=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?
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