Question

A body mass $M$ at rest is struck by a moving body of mass $m$. Show that the fraction of the initial kinetic energy of moving mass $m$ transferred to the strucked body is $4mM/(m+M{)}^2$.

Answer 1

• Let the body of mass m moves with velocity u towards mass M at rest.
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• initial K.E. of the moving body

$E_1=\frac{1}{2}m{u}_1^2$

= $\frac{1}{2}m{u}^2$

After collision,

$v_2$ = $\frac{2m_1{u}_1}{m_1+m_2}$ + $\frac{(m_2-m_1)u_2}{m_1+m_2}$

=$\frac{2mu}{m+M}+0$

= $\frac{2mu}{m+M}$

Kinetic energy of the mass M,

$E_2=\frac{1}{2}{m}_2{v}_2^2$

$=\frac{1}{2}M(\frac{2mu}{M+m}{)}^2$

= $\frac{2M{m}^2{u}^2}{(M+m{)}^2}$

Hence, fraction of K.E. transferred. ,

$\frac{E_2}{E_1}=\frac{2M{m}^2{u}^2}{(m+M{)}^2(\frac{1}{2}m{u}^2)}$

= $\frac{4Mm}{(M+m{)}^2}$