Question

Three blocks $A, B$ and $C$ are lying on a smooth horizontal surface, as shown in the figure. $A$ and $B$ have equal masses, $m$ while $C$ has mass $M$. Block $A$ is given an brutal speed $v$ towards $B$ due to which it collides with $B$ perfectly inelastically. The combined mass collides with $C$, also perfectly inelastically $\frac{5}{6}$ th of the initial kinetic energy is lost in whole process. What is value of $M/m$ ?

• A. 4
• B. 5
• C. 3
• D. 2

Answer 1

The Correct Option is A

To find the value of $M/m$, we need to analyze the problem using the principles of conservation of momentum and kinetic energy loss in inelastic collisions.

1. Initially, block $A$ with mass $m$ is moving with velocity $v$, and blocks $B$ and $C$ are at rest.
2. First Collision (A and B):
   • Since the collision between $A$ and $B$ is perfectly inelastic, the two blocks stick together after the collision.
   • By conservation of momentum, the velocity $u$ of the combined mass after the collision can be given by:
   • $m \cdot v = (m + m) \cdot u \Rightarrow u = \frac{v}{2}$
3. Second Collision (Combined mass of A & B with C):
   • Again, the collision is perfectly inelastic, so all three blocks move together as a single unit.
   • Let the final velocity of the combined mass after this collision be $V$.
   • Using conservation of momentum:
   • $(2m) \cdot \frac{v}{2} = (2m + M) \cdot V \Rightarrow v \cdot m = (2m + M) \cdot V$
   • So, $ V = \frac{v \cdot m}{2m + M} $
4. Energy Analysis:
   • Initial kinetic energy: $ KE_{\text{initial}} = \frac{1}{2} m v^2 $
   • Final kinetic energy: $ KE_{\text{final}} = \frac{1}{2} \cdot (2m + M) \cdot V^2 $
   • Since $\frac{5}{6}$ th of the initial kinetic energy is lost, $\frac{1}{6}$ th remains:
   • $\frac{1}{2} \cdot (2m + M) \cdot \left(\frac{v \cdot m}{2m + M}\right)^2 = \frac{1}{6} \cdot \frac{1}{2} m v^2 $
   • This results in: $ (2m + M) \cdot \frac{m^2 \cdot v^2}{(2m + M)^2} = \frac{m v^2}{6} $
   • Simplifying gives: $\frac{m^2}{2m + M} = \frac{m}{6}$
   • Cross-multiplying and simplifying:
   • $6m^2 = m(2m + M)$
   • Which results in: $4m^2 = mM \; \Rightarrow \; M = 4m $
5. Thus, the value of $M/m$ is 4, which matches the correct option.

Therefore, the correct answer is 4.