Question

A body initially at rest breaks up into two pieces of masses $M$ and $3M$ and move with a total kinetic energy of $E$, then the kinetic energy of the piece of mass $M$ is

• A. $\frac{3E}{4}$
• B. $\frac{E}{4}$
• C. $\frac{2E}{3}$
• D. $\frac{E}{3}$
• E. $\frac{E}{2}$

Hint:

In explosion problems: - Momentum is conserved - Use ratio of velocities to split kinetic energy efficiently

Answer 1

The Correct Option is A

Concept: Since the body was initially at rest, total momentum is conserved: \[ M v_1 = 3M v_2 \] Also, kinetic energy: \[ K = \frac{1}{2}mv^2 \]

Step 1: Apply conservation of momentum.
\[ Mv_1 = 3Mv_2 \Rightarrow v_1 = 3v_2 \]

Step 2: Write kinetic energies.
\[ K_1 = \frac{1}{2}M(3v_2)^2 = \frac{9}{2}Mv_2^2 \] \[ K_2 = \frac{1}{2}(3M)v_2^2 = \frac{3}{2}Mv_2^2 \]

Step 3: Find ratio of energies.
\[ K_1 : K_2 = 9 : 3 = 3 : 1 \]

Step 4: Use total energy.
\[ K_1 + K_2 = E \] \[ 3x + x = 4x = E \Rightarrow x = \frac{E}{4} \]

Step 5: Find $K_1$.
\[ K_1 = 3x = \frac{3E}{4} \]