A bullet of mass $m$ hits a stationary block of mass $M$ elastically. The transfer of energy is the maximum, w

Question

A bullet of mass $m$ hits a stationary block of mass $M$ elastically. The transfer of energy is the maximum, when:

Accepted Answer

1. **Analyze Elastic Collision:** In a head-on elastic collision between a projectile of mass $m_1$ (bullet, $m$) and a stationary target of mass $m_2$ (block, $M$), kinetic energy is conserved.
2. **Energy Transfer:** The fractional kinetic energy transferred from the projectile to the target is given by:
   $$ \frac{K_{block}}{K_{bullet}} = \frac{4m_1 m_2}{(m_1 + m_2)^2} = \frac{4mM}{(m + M)^2} $$
3. **Maximize Transfer:** This fraction is maximum (equal to 1 or 100%) when the numerator equals the denominator, i.e., $4mM = (m + M)^2$, which simplifies to $(M - m)^2 = 0$, implying $M = m$.
4. **NCERT Reference:** This corresponds to **Case I** in the text: 'If the two masses are equal... The first mass comes to rest and pushes off the second mass with its initial speed on collision.' Thus, the entire kinetic energy is transferred [Class 11 Physics, Ch 5, Sec 5.11.2, Case I].

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