Question

A body of mass 'm' moving with a velocity of 'v' collides head on with another body of mass '2m' at rest. If the coefficient of restitution between the two bodies is 'e', then the ratio of the velocities of the two bodies after collision is

• A. \(\frac{1+e}{1-2e}\)
• B. \(\frac{1+2e}{1-e}\)
• C. \(\frac{1-e}{1+2e}\)
• D. \(\frac{1-2e}{1+e}\)

Hint:

For any 1D collision problem, the process is standard: write the momentum conservation equation and the coefficient of restitution equation. This always gives you two equations to solve for the two unknown final velocities. Be careful with signs.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem involves a one-dimensional (head-on) collision. To solve it, we need to apply two fundamental principles: the conservation of linear momentum and the definition of the coefficient of restitution.
Step 2: Setting up the Equations:
Let the initial velocities be \(u_1 = v\) and \(u_2 = 0\).
Let the masses be \(m_1 = m\) and \(m_2 = 2m\).
Let the final velocities be \(v_1\) and \(v_2\).
Equation from Conservation of Momentum:
\[ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \] \[ m(v) + 2m(0) = m(v_1) + 2m(v_2) \] Dividing by \(m\):
\[ v = v_1 + 2v_2 \quad \cdots(1) \] Equation from Coefficient of Restitution (e):
\[ e = \frac{\text{velocity of separation}}{\text{velocity of approach}} = \frac{v_2 - v_1}{u_1 - u_2} \] \[ e = \frac{v_2 - v_1}{v - 0} \] \[ ev = v_2 - v_1 \quad \cdots(2) \] Step 3: Solving the Equations:
We have a system of two linear equations with two unknowns, \(v_1\) and \(v_2\). We need to find the ratio \(v_1/v_2\).
From equation (2), we can express \(v_1\) as \(v_1 = v_2 - ev\).
Substitute this expression for \(v_1\) into equation (1):
\[ v = (v_2 - ev) + 2v_2 \] \[ v = 3v_2 - ev \] \[ v + ev = 3v_2 \implies v(1+e) = 3v_2 \] \[ v_2 = \frac{v(1+e)}{3} \] Now, substitute this result for \(v_2\) back into the expression for \(v_1\):
\[ v_1 = v_2 - ev = \frac{v(1+e)}{3} - ev \] \[ v_1 = \frac{v(1+e) - 3ev}{3} = \frac{v + ve - 3ev}{3} = \frac{v - 2ev}{3} = \frac{v(1-2e)}{3} \] Step 4: Finding the Ratio:
Now we can find the ratio of the velocities, \(\frac{v_1}{v_2}\).
\[ \frac{v_1}{v_2} = \frac{\frac{v(1-2e)}{3}}{\frac{v(1+e)}{3}} \] \[ \frac{v_1}{v_2} = \frac{1-2e}{1+e} \] This matches option (D).