Question

Help! Elastic Collision.

Two objects, one initially at rest, undergo a one-dimensional elastic collision. If half the kinetic energy of the initially moving object is transferred to the other object, what is the ratio of their masses?

Answer 1

assume that \[v_1=2v\] and \[v_2=0\] before the collision and \[m_1=m\] and \[m_2=M\] we need to find an equation between m and M.
Before the collision \[KE_1=1/2m(2v)^2=2mv^2\] and \[KE_2=0\]Also; \[P_1=2mv\] and \[P_2=0\] After collision (because it is an elastic collision, KE doesn't change.);
(and let us take m1(m) has v1 and m2(M) has v2 velocities after collision) we can write these equations;
[notice that both will have \[mv^2\] kinetic energy]
\[mv^2=1/2mv_1^2\] and we have \[v_1=\sqrt{2}v\]

Also for m2 ;\[mv^2=1/2Mv_2^2\] and we have \[v_2=v \sqrt{2m/M}\]

and finally we know that final momentum should be equal to initial momentum which was equal to P1= 2mv. so;
\[2mv=Mv_2-mv_1\] you directly plug in v1 and v2 from above and you should get such and equaition \[m(\sqrt{2}-1)^2=M\]

i hope i haven't made any mistake.