Question

State and prove the law of conservation of linear momentum.

Accepted Answer

The law of conservation states that in the absence of an external force, the algebraic sum of the momentum after collision is numerically equal to the algebraic sum of the momentum before collision.

Let us prove the law of conservation of linear momentum with the following illustration:

Let two bodies A and B having masses $m_{1}$ and $m_{2}$ move with initial velocity $u_{1}$ and $u_{2}$ in a straight line. Let the velocity of the first body be higher than that of the second body. i.e., $u_{1} > u_{2}$ . During an interval of time $t$ second, they tend to have a collision. After the impact, both of them move along the same straight line with a velocity $v_{1}$ and $v_{2}$ respectively.

Force on body B due to A,
$F_{B} = \frac{m_{2} (v_{2} - u_{2})}{t}$
Force on body A due to B,
$F_{A} = \frac{m_{1} (v_{1} - u_{1})}{t}$

By Newton’s third law of motion,
Action force $=$ Reaction force
$F_{A} = - F_{B}$
$\Rightarrow \frac{m_{1} (v_{1} - u_{1})}{t} = - \frac{m_{2} (v_{2} - u_{2})}{t} \Rightarrow m_{1} v_{1} + m_{2} v_{2} = m_{1} u_{1} + m_{2} u_{2}$

State and prove the law of conservation of linear momentum.

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