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2D Collision Example

two bodies of...

Question

two bodies of equal masses moving with different velocities in the same straight line undergoes elastic collision. show that they exchange their velocities after collision?

which derivation should we mention elastic collision in one dimension or elastic collision in two dimensions

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Solution

Let the body of mass $m_{1}$ moving with velocity $u_{1}$ moves with velocity $v_{2}$ after collision with the body of mass $m_{2}$ which was moving with velocity $u_{2}$ and which after collision attains a velocity $v_{2}$ .
$∴ v_{1} \frac{(m_{1} - m_{2})}{m_{1} + m_{2}} u_{1} + \frac{2 m_{2}}{m_{1} + m_{2}} u_{2}$
And,
$v_{2} = \frac{2 m_{1}}{m_{1} + m_{2}} u_{1} - \frac{(m_{1} + m_{2})}{m_{1} + m_{2}} u_{2}$
When $m_{1} = m_{2} = m$
$v_{1} = \frac{0}{2 m} u_{1} + \frac{2 m}{2 m} u_{2} = u_{2}$
And
$v_{2} = \frac{2 m}{2 m} u_{1} - \frac{0}{2 m} u_{2} = u_{1}$
Thus, the bodies exchange speeds after collision.

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