Principle of Conservation of Linear Momentum and its Applications
If the net external force acting on a system of bodies is zero, then the momentum of the system remains constant. This is the basic principle of conservation of linear momentum.
According to Newton’s second law
If net force (or the vector sum of all forces) on system of particle is equal to zero, the vector sum of linear momentum of all particles remains conserved.
Consider a system of two bodies on which no external force acts. The bodies can mutually interact with each other. Due to the mutual interaction of the bodies the momentum of the individual bodies may increase or decrease according to the situation, but the momentum of the system will always be conserved, as long as there is no external net force acting on it.
Thus, if \(\vec{p}_{1}\) and \(\vec{p}_{2}\) are momentum of the two bodies at any instant, then in absence of external force
\(\vec{p}_{1}+\vec{p}_{2}\) = constant ……………. (3)
If due to mutual interaction, the momentum of two bodies becomes \(\overrightarrow{p_{1}^{\prime}}\) and \(\overrightarrow{p_{2}^{\prime}}\) respectively, then according to principle of conservation of momentum
\(\overrightarrow{p_{1}}+\overrightarrow{p_{2}}=\overrightarrow{p_{1}^{\prime}}+\overrightarrow{p_{2}^{\prime}}\)
or \(m_{1} \overrightarrow{u_{1}}+m_{2} \overrightarrow{u_{2}}=m_{1} \overrightarrow{v_{1}}+m_{2} \overrightarrow{v_{2}}\) ……………. (4)
Where \(\overrightarrow{u_{1}}\) and \(\overrightarrow{u_{2}}\) are initial velocities of the two bodies of masses m1 and m2 and \(\overrightarrow{v_{1}}\) and \(\overrightarrow{v_{2}}\) are their final velocities.
Therefore, the principle of conservation of linear momentum may also be stated as follows :
For an isolated system (a system on which no external force acts), the initial momentum of the system A is equal to the final momentum of the system.
Practical Applications of the Principle of Conservation of Momentum
1. Recoiling a Gun : Lets consider the gun and bullet in its barrel as an isolated system. In the beginning when bullet is not fired both the gun and the bullet are at rest. So the momentum before firing is zero
or \(\overrightarrow{p_{c}}\) = 0
Now when the bullet is fired, it moves , in the forward direction and gun recoils back in the opposite direction.
Let mb be the mass and vb be the velocity of the bullet and mg and vg be the mass and velocity of the gun after firing.
Total momentum of the system after the firing would be
\(\overrightarrow{p_{f}}=m_{b} \overrightarrow{v_{b}}+m_{g} \overrightarrow{v_{g}}\)
Since, no external forces are acting on the system, we can apply the law of conservation of linear momentum therefore,
Total momentum of gun and bullet before firing
= Total momentum of gun and bullet after firing
\(0=m_{b} \overrightarrow{v_{b}}+m_{g} \vec{v}_{g}\)
or \(\overrightarrow{v_{g}}=-\frac{m_{b} \vec{v}_{b}}{m_{g}}\)
The negative sign shows that \(\vec{v}_{g}\) and \(\vec{v}_{b}\) are in opposite directions i.e., as the bullet moves forward, then the gun will move in backward direction. The backward motion of the gun is called recoil of the gun.
2. While firing a bullet, the gun must be held tight to the shoulder : This would save hurting the shoulder of the man who fires the gun as the recoil velocity of the gun. If the gun is held tight to the shoulder, then the gun and the body of the man recoil as one system. As the total mass is quite large, the recoil velocity will be very small and the shoulder of the man will not get hurt.
3. Rockets works on the principle of conservation of momentum : The rocket’s fuel burns and pushes the exhaust gases downwards, due to this the rocket gets pushed upwards. Motorboats also work on the same principle, it pushes the water backwards and gets pushed forward in reaction to conserve momentum.
Second Law of Motion is the real law of Motion :
(A) First law is contained in the second law :
According to Newton’s second law of motion, the force acting on a body is given by :
Thus there is no force is applied on the body then the body at rest will remains at rest and a body in uniform motion will continue to move uniformly along the same straight path. Hence first law of motion is contained in the second law.
