Anibphysicsproblemaboutkineticenergyandmomentum!Showthaninanelastic,head-oncollisionofaparticleofmassmwithastationaryparticleofmassMis(4mM)/(m+M)^2.That'sthequestion.

Question

An ib physics problem about kinetic energy and momentum!
Show than in an elastic,head-on collision of a particle of mass m with a stationary particle of mass M is (4mM)/(m+M)^2.That's the question.

Answer 1

Use energy conservation and momentum conservation laws.
Before collision, the velocity of m is v, and M is 0. Assume that after collision the velocity of m is u and that of M is w, then
momentum conservation:
mv=mu+Mw
kinetic energy conservation (because collision is elastic)
(1/2)mv²＝(1/2)mu²+(1/2)Mw²
From the first equation we find
u=v-(M/m)w
Substituting into the second equation, and solve for w, we find,
w=2mv/(m+M)
or
w/v=2m/(m+M)
Therefore, the ratio between the kinetic energy of M after the collision and the total kinetic energy (which is equal to the kinetic energy of m before the collision because of energy conservation) is,
(1/2)Mw²/[(1/2)mv²]=(M/m)(w/v)²=(M/m)[4m²/(m+M)²]=4mM/(m+M)²