Question

State and verify law of conservation of momentum.

Answer 1

Law of conservation of momentum:

The law of conservation of momentum asserts that in an isolated system with two or more objects working on one other, the overall momentum of the system remains constant until an external force is added. This also implies that the total momentum of an isolated system prior to isolation is equal to the total momentum of the isolated system subsequent to isolation.

Verification:

In order to verify law of conservation of momentum, let's take two bodies A and B having mass ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ respectively. Let the initial velocities of both bodies be u₁ and u₂, after the elastic collision let's assume that the final velocities of both bodies be ${\mathrm{v}}_1$ and ${\mathrm{v}}_2$ respectively. As we know that the momentum $‘\mathrm{p}’$ of a body having mass $‘\mathrm{m}’$ and velocity $‘\mathrm{v}’$ is;

$p=mv\left(1\right)$

We can write change in momentum by using equation (1);

$Changeinmomentum=Mass(finalvelocity-initialvelocity)\left(2\right)$

Change in momentum for body A will be;

$ChangeinmomentumofbodyA=m_1(v_1-u_1)$

Similarly change in momentum for body B will be;

$ChangeinmomentumforbodyB=m_2(v_2-u_2)$

Now, we can claim that any action has an equal and opposite response, according to Newton's third law of motion. That is;

Force applied by B on A = Force applied by A on B

$F_{AB}=-F_{BA}\left(3\right)$

Here negative sign shows that both forces occur in opposite direction.

The force F applied to a body is equal to the rate of change of momentum p of the body with respect to time, according to Newton's second law of motion.

$$\begin{gathered} F=\frac{dp}{dt} \\ F=\frac{∆p}{dt}\left(4\right) \end{gathered}$$

By using (4), we have;

$$\begin{gathered} F_{AB}=\frac{ChangeinmomentumofbodyA}{Timetaken} \\ =\frac{m_1(v_1-u_1)}{t} \end{gathered}$$

Similarly,

$$\begin{gathered} F_{BA}=\frac{ChangeinmomentumofbodyB}{Timetaken} \\ =\frac{m_2(v_2-u_2)}{t} \end{gathered}$$

Now, by using equation (3), we get;

$$\begin{gathered} \Rightarrow \frac{m_1(v_1-u_1)}{t}=-\frac{m_2(v_2-u_2)}{t} \\ \Rightarrow m_1(v_1-u_1)=-m_2(v_2-u_2) \\ \Rightarrow m_1{v}_1-m_1{u}_1=-m_2{v}_2+m_2{u}_2 \\ \Rightarrow m_1{v}_1+m_2{v}_2=m_1{u}_1+m_2{u}_2 \end{gathered}$$

The total momentum of the system after collision equals the total momentum of the system before contact, according to the preceding equation. Hence, law of conservation of momentum is verified.