Question

Prove the Law of Conservation of Momentum [5 MARKS]

Answer 1

Explanation: 2 Marks
Proof: 3 Marks

The law of momentum conservation can be stated as follows:
For a collision occurring between object 1 and object 2 in an isolated sytem. the total momentum of the two objects before the collision is equal to the total momentum of the two objects after the collision. That is, the momentum lost by object 1 is equal to the momentum gained by object 2.

Proving Law of Conservation of Momentum

Suppose two objects (two balls A and B, say) of masses $m_A$ and $m_B$ are travelling in the same direction along a straight line at different velocities $u_A$ and $u_B$. And there are no other external unbalanced forces acting on them.

During collision which lasts for a time t, the ball A exerts a force $F_{AB}$ on ball B and the ball B exerts a force $F_{BA}$ on ball A.

Suppose $V_A$ and $V_B$ are the velocities of the two balls A and B after the collision respectively

then the rate of change in momentum in body A due to $F_{BA}$ is

$F_{BA}=\frac{m_A{v}_A-m_A{u}_A}{t}$

the rate of change of momentum in body B due to $F_{AB}$ is

$F_{AB}=\frac{m_B{v}_B-m_B{u}_B}{t}$

from the newton's third law of motion

$F_{AB}=-F_{BA}$

$\frac{m_B{v}_B-m_B{u}_B}{t}=-\frac{m_A{v}_A-m_A{u}_A}{t}$

$\Rightarrow m_A{u}_A+m_B{u}_B=m_A{v}_A+m_B{v}_B$

From this it is clear that the sum of initial momenta of objects before collision is equal to the sum of final momenta of object after collision, and thus it is the case of conservation of momentum in the absence of external force.