Question

State and verify the law of conservation of momentum.

Accepted Answer

According to the law of conservation of linear momentum, in the absence of an external force, the total momentum of all the bodies of a system remains constant (conserved).

Proof:

Let the two bodies A and B have masses $m_{1}$ and $m_{2}$ and their initial velocities be $u_{1}$ and $u_{2}$ respectively.

According to the definition of momentum,

momentum of body $A = m_{1} u_{1}$

momentum of body $B = m_{2} u_{2}$

Initial momentum of the system, $p_{i} = m_{1} u_{1} + m_{2} u_{2}$

Let the two bodies collide. After collision their velocities be $v_{1}$ and $v_{2}$ respectively.

Final momentum of the system, $p_{f} = m_{1} v_{1} + m_{2} v_{2}$

Body $A$ exerts a force of action $F_{A B}$ on body $B$ .

Body $B$ exerts a force of reaction $F_{B A}$ on body $A$ .

Let the collision last for $t$ seconds.

_{$F_{A B} = Rate of change of momentum of body A$}

_{$F_{A B} = \frac{m_{1} v_{1} - m_{1} u_{1}}{t} = \frac{m_{1} (v_{1} - u_{1})}{t}$}

$F_{BA} = Rate of change of momentum of body B$

$F_{BA} = \frac{m_{2} v_{2} - m_{2} u_{2}}{t} = \frac{m_{2} (v_{2} - u_{2})}{t}$

From, Newton's third law of motion

$action = - reaction$

$F_{A B} = - F_{B A}$

$\frac{m_{1} (v_{1} - u_{1})}{t} = - \frac{m_{2} (v_{2} - u_{2})}{t}$

$m_{1} v_{1} - m_{1} u_{1} = - m_{2} v_{2} + m_{2} u_{2}$
$m_{1} v_{1} + m_{2} v_{2} = m_{1} u_{1} + m_{2} u_{2}$
Therefore, Momemtum of the system after collision=Momentum of the system before collision.

This is the law of conservation of momentum.

State and verify the law of conservation of momentum.

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