A mass $m$ (moving along the $x$-axis) with velocity $v$ collides and sticks to a mass of $3m$ moving vertical

Question

A mass $m$ (moving along the $x$-axis) with velocity $v$ collides and sticks to a mass of $3m$ moving vertically upward (along the $y$-axis) with velocity $2v$. The final velocity of the combination is:

Accepted Answer

1. **Identify the System:** This is a completely inelastic collision in two dimensions where the two masses stick together. Linear momentum is conserved.
2. **Initial Momentum:**
   - Mass 1 ($m$): $\vec{p}_1 = m v \hat{i}$
   - Mass 2 ($3m$): $\vec{p}_2 = 3m (2v) \hat{j} = 6mv \hat{j}$
   - Total Initial Momentum: $\vec{P}_{initial} = mv \hat{i} + 6mv \hat{j}$
3. **Final Momentum:**
   - Combined Mass: $M = m + 3m = 4m$
   - Final Velocity: $\vec{v}_f$
   - Total Final Momentum: $\vec{P}_{final} = (4m) \vec{v}_f$
4. **Apply Conservation of Momentum:**
   $$ \vec{P}_{initial} = \vec{P}_{final} $$
   $$ mv \hat{i} + 6mv \hat{j} = 4m \vec{v}_f $$
   $$ \vec{v}_f = \frac{mv \hat{i} + 6mv \hat{j}}{4m} = \frac{1}{4}v \hat{i} + \frac{6}{4}v \hat{j} = \frac{1}{4}v \hat{i} + \frac{3}{2}v \hat{j} $$
   [Class 11 Physics, Ch 6, Sec 5.11.3]

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