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Step-by-Step Solution
Step 1: Identify the Given Information
• A particle of mass $m$ moves initially with velocity $u \hat{i}$.
• It collides elastically with another particle of mass $3m$ at rest.
• After collision, the first particle moves with velocity $v \hat{j}$.
• We need to find the magnitude of $v$ in terms of $u$.
Step 2: Apply Conservation of Momentum
Before collision, total momentum is
$ m \, u \hat{i} $ (since the $3m$ mass is at rest).
Let the velocity of the $3m$ mass after collision be $\overrightarrow{v'}$. Then the total momentum after collision is
$ m \, v \hat{j} + 3m \, \overrightarrow{v'}. $
Hence, conservation of momentum gives
$ m \, u \hat{i} = m \, v \hat{j} + 3 m \, \overrightarrow{v'}. $
Rearranging for $\overrightarrow{v'}$:
$ 3m \, \overrightarrow{v'} = m \, u \hat{i} - m \, v \hat{j}, $
$ \overrightarrow{v'} = \frac{u \hat{i} - v \hat{j}}{3}. $
The magnitude of $\overrightarrow{v'}$ is
$ \left|\overrightarrow{v'}\right|
= \frac{\sqrt{u^2 + v^2}}{3}. $
Step 3: Apply Conservation of Kinetic Energy
Because the collision is perfectly elastic, the total kinetic energy before and after collision must be the same:
$ \frac{1}{2} m u^2 + \frac{1}{2} \cdot 3m \cdot 0^2
= \frac{1}{2} m v^2 + \frac{1}{2} \cdot 3m \left|\overrightarrow{v'}\right|^2. $
Simplify:
$ \frac{1}{2} m u^2 = \frac{1}{2} m v^2 + \frac{3}{2} m \left(\left|\overrightarrow{v'}\right|^2\right). $
Divide through by $\frac{1}{2}m$:
$ u^2 = v^2 + 3 \left|\overrightarrow{v'}\right|^2. $
But from Step 2, we have
$ \left|\overrightarrow{v'}\right|^2
= \frac{u^2 + v^2}{9}. $
Substitute this into the kinetic energy equation:
$ u^2 = v^2 + 3 \left(\frac{u^2 + v^2}{9}\right). $
Simplify:
$ u^2 = v^2 + \frac{u^2 + v^2}{3}, $
$ 3u^2 = 3v^2 + (u^2 + v^2). $
$ 3u^2 = 4v^2 + u^2. $
$ 2u^2 = 4v^2. $
$ \frac{2u^2}{4} = v^2. $
$ v^2 = \frac{u^2}{2}. $
Taking the positive root (since velocity magnitude is positive),
$ v = \frac{u}{\sqrt{2}}. $
Step 4: Final Answer
Thus, the speed of the mass $m$ after collision is
$ \displaystyle \frac{u}{\sqrt{2}}. $
Below is the referenced solution image (unchanged):
