Question

A mass \( m \) is connected to a massless spring of spring constant \( k \), which is fixed to a wall. Another mass \( 2m \), having kinetic energy \( E \), collides collinearly with the mass \( m \) completely inelastically (see figure). The entire setup is placed on a frictionless floor. The maximum compression of the spring is: 

• A. \( \sqrt{\frac{4E}{3k}} \)
• B. \( \sqrt{\frac{E}{3k}} \)
• C. \( \sqrt{\frac{E}{5k}} \)
• D. \( \sqrt{\frac{E}{7k}} \)

Hint:

For inelastic collisions followed by spring compression, use conservation of momentum during collision and conservation of energy after collision.

Answer 1

The Correct Option is A

Step 1: Apply conservation of momentum.
Before collision: \[ p_{\text{initial}} = \sqrt{4mE}. \] After inelastic collision (masses stick together): total mass = \( 3m \), and velocity \( v \): \[ 3mv = \sqrt{4mE} \Rightarrow v = \sqrt{\frac{4E}{9m}}. \] Step 2: Apply conservation of energy after collision.
Kinetic energy after collision is converted into spring potential energy at maximum compression: \[ \frac{1}{2}(3m)v^2 = \frac{1}{2}k x^2. \] Substitute \( v^2 = \frac{4E}{9m} \): \[ \frac{1}{2}(3m)\left(\frac{4E}{9m}\right) = \frac{1}{2}k x^2 \Rightarrow \frac{2E}{3} = \frac{1}{2}k x^2. \] \[ x = \sqrt{\frac{E}{3k}}. \] Step 3: Final Answer.
Maximum compression of the spring is \( \sqrt{\frac{E}{3k}}. \)