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Step-by-Step Solution
Step 1: Identify the Given Parameters
• Let the mass of the first ball be $m_1 = m$.
• Let the mass of the second ball be $m_2 = 2m$.
• Initial velocity of the first ball, $u_1 = 2\ \text{m/s}$.
• Initial velocity of the second ball, $u_2 = 0\ \text{m/s}$.
• Coefficient of restitution, $e = 0.5$.
We want the velocities $v_1$ and $v_2$ after collision.
Step 2: Apply Conservation of Linear Momentum
Conservation of momentum states that the total momentum before collision equals the total momentum after collision. Mathematically:
$ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 $
Substitute the values:
$ m \times 2 + (2m) \times 0 = m \times v_1 + (2m) \times v_2 $
$ 2m = m\,v_1 + 2m\,v_2 $
Divide both sides by $m$:
$ 2 = v_1 + 2\,v_2 \quad \text{...(i)}$
Step 3: Apply the Definition of Coefficient of Restitution
The coefficient of restitution $e$ is given by:
$ e = \frac{v_2 - v_1}{u_1 - u_2} $
Substitute the known values $e = 0.5$, $u_1 = 2$, and $u_2 = 0$:
$ 0.5 = \frac{v_2 - v_1}{2 - 0} \quad \Longrightarrow \quad 0.5 \times 2 = v_2 - v_1 $
$ 1 = v_2 - v_1 \quad \text{...(ii)} $
Step 4: Solve the System of Equations
We now have two simultaneous equations:
(i) $ 2 = v_1 + 2v_2 $
(ii) $ 1 = v_2 - v_1 $
From equation (ii), $v_2 = 1 + v_1$. Substitute this into (i):
$ 2 = v_1 + 2(1 + v_1) = v_1 + 2 + 2v_1 = 3v_1 + 2 $
$ 3v_1 = 0 \quad \Longrightarrow \quad v_1 = 0 $
Then from (ii),
$ 1 = v_2 - 0 \quad \Longrightarrow \quad v_2 = 1 $
Step 5: State the Final Velocities
Thus, the velocity of the first ball after collision is $ v_1 = 0\ \text{m/s} $, and the velocity of the second ball is $ v_2 = 1\ \text{m/s} $.
