Question

A ball of mass 200 g moving with certain velocity collides with another ball of mass 600 g at rest. If the coefficient of restitution is 0.6, the ratio of the velocity of 600 g ball after collision and the velocity of 200 g ball before collision is:

• A. \( 1:2 \)
• B. \( 3:5 \)
• C. \( 2:3 \)
• D. \(\mathbf{2:5}\)

Hint:

- The coefficient of restitution (e) relates velocities before and after collision. - Momentum conservation is always applied in collision problems. - Use \( e = \frac{v_2 - v_1}{u_1 - u_2} \) and momentum equation together to solve.

Answer 1

The Correct Option is D

Step 1: Understanding the given problem We apply the equation for the coefficient of restitution (e), which is given by: \[ e = \frac{v_2 - v_1}{u_1 - u_2} \] where: - \( u_1 \) and \( u_2 \) are initial velocities of the 200 g and 600 g balls respectively. - \( v_1 \) and \( v_2 \) are final velocities after the collision. - Given that the 600 g ball is initially at rest, \( u_2 = 0 \). - The coefficient of restitution is given as \( e = 0.6 \).

 Step 2: Applying the restitution formula Since \( u_2 = 0 \), the equation simplifies to: \[ 0.6 = \frac{v_2 - v_1}{u_1} \] Rearranging: \[ v_2 - v_1 = 0.6 u_1 \]

 Step 3: Applying momentum conservation Momentum before and after collision must be equal: \[ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \] Substituting given values: \[ (0.2) u_1 + (0.6) (0) = (0.2) v_1 + (0.6) v_2 \] \[ 0.2 u_1 = 0.2 v_1 + 0.6 v_2 \] 

Step 4: Solving for \( v_1 \) and \( v_2 \) From the restitution equation: \[ v_1 = v_2 - 0.6 u_1 \] Substituting in the momentum equation: \[ 0.2 u_1 = 0.2 (v_2 - 0.6 u_1) + 0.6 v_2 \] \[ 0.2 u_1 = 0.2 v_2 - 0.12 u_1 + 0.6 v_2 \] \[ 0.2 u_1 + 0.12 u_1 = 0.2 v_2 + 0.6 v_2 \] \[ 0.32 u_1 = 0.8 v_2 \] Solving for \( v_2 \): \[ v_2 = \frac{0.32}{0.8} u_1 = 0.4 u_1 \] Now, using \( v_1 = v_2 - 0.6 u_1 \): \[ v_1 = 0.4 u_1 - 0.6 u_1 = -0.2 u_1 \]

 Step 5: Finding the ratio \[ \frac{v_2}{u_1} = \frac{0.4 u_1}{u_1} = 0.4 \] Expressing as a ratio: \[ \frac{v_2}{u_1} = \frac{2}{5} \] Thus, the correct ratio is: \[ \mathbf{2:5} \]