a body of mass M at rest is struck by a moving body of mass m prove that the - Physics - Work Energy And Power

Question

a body of mass M at rest is struck by a moving body of mass m
prove that the fraction of initial K E of mass transferred to the
struck body is 4Mm/(m+M)^2

Rajendra Kumar · Accepted Answer

Let body of mass m moves with velocity u towards mass M at rest.

Initial K.E. of the moving body

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

After collision,

$v_{2} = \frac{2 m_{1} u_{1}}{m_{1} + m_{2}} + \frac{(m_{2} - m_{1}) u_{2}}{m_{1} + m_{2}} = \frac{2 m u}{m + M} + 0 = \frac{2 m u}{m + M}$

Kinetic energy of the mass M,

$E_{2} = \frac{1}{2} m_{2} v_{2}^{2} = \frac{1}{2} M {(\frac{2 m u}{M + m})}^{2} = \frac{2 M m^{2} u^{2}}{{(M + m)}^{2}}$

Hence, fraction of K.E. transferred.,

$\frac{E_{2}}{E_{1}} = \frac{2 M m^{2} u^{2}}{{(m + M)}^{2} (\frac{1}{2} m u^{2})} = \frac{4 M m}{{(M + m)}^{2}}$

a body of mass M at rest is struck by a moving body of mass m. prove that the fraction of initial K.E of mass transferred to the struck body is 4Mm/(m+M)^2