Two particles are moving towards each other along a line joining them, so that their centre of mass does not move. After elastic collision between them,

their centre of mass will move

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their velocities will not change

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their speeds will not change

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their velocities will not change only if they have equal mass

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Solution

The correct option is B their speeds will not change
As the center of mass is at rest therefore it is a center of momentum frame, hence total momentum will be zero.
$m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2} = 0$ ,
or $u_{1} = - m_{2} u_{2} / m_{1}, v_{1} = - m_{2} v_{2} / m_{1}$ ,
Now, it is an elastic collision hence kinetic energy remains constant.
$m_{1} u_{1}^{2} + m_{2} u_{2}^{2} = m_{1} v_{1}^{2} + m_{2} v_{2}^{2}$ ,
or $(m_{2} u_{2})^{2} / m_{1} + (m_{2} u_{2})^{2} / m_{2} = (m_{1} v_{1})^{2} / m_{1} + (m_{2} v_{2})^{2} / m_{2}$ ,
or $(m_{1} + m_{2}) (m_{2} u_{2}^{2}) = (m_{1} + m_{2}) (m_{2} v_{2}^{2})$ ,
or $u_{2} = - v_{2}$ ,
Similarly, $u_{1} = - v_{1}$ ,
Therefore the speeds will not change.

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