Two blocks of mass m and M are lying on a smooth table. A spring is attached with the block of mass M (see fig). Block of mass m is given a velocity towards the other block. Find the maximum value of $\frac{M}{m}$ for which the kinetic energy of the system will never fall below one third of the initial kinetic energy imparted to the block of mass m

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Detailed Solution

KE of the system is minimum when compression in the spring is maximum. This happens when both the blocks have same velocity (Say V). If initial velocity of mass m is u then–
$(M + m) V = mu \Rightarrow V = \frac{mu}{M + m}$

According to the problem
$\frac{1}{2} (M + m) {(\frac{mu}{M + m})}^{2} \geq \frac{1}{3} \cdot \frac{1}{2} {mu}^{2}$
$\Rightarrow \frac{m}{M + m} \geq \frac{1}{3} \Rightarrow \frac{M}{m} \leq 2$

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