# Newton's Third Law of Motion

- Conservation of momentum
- Practical applications of conservation of momentum

Newton’s Third Law of Motion states that there is an equal and opposite reaction to every action.

**Explanation:** Whenever a force is applied on a body, that body also applies the same force and in opposite direction. The force with which you hit a wall, the wall hits you with the same force.

**Examples**

**Walking of a person:**Newton’s Third Law of Motion is at play when a person walks. During walking, a person pushes the ground in backward direction. In the reaction to that, the ground pushes the person with equal magnitude of force but in opposite direction. This enables him to move in forward direction against the push.**Recoil of gun:**When a bullet is fired from a gun, the bullet pushes the gun in opposite direction, with equal magnitude of force. This results in gunman feeling a backward push from the butt of gun. This is called*recoil of gun.***Propulsion of a boat in forward direction:**A sailor pushes water with oar in backward direction, and water reacts by pushing the oar in forward direction. Consequently, the boat is pushed in forward direction. Forces applied by oar and water are of equal magnitude but in opposite directions.

## Conservation of Momentum:

**Law of Conservation of Momentum:** The sum of momenta of two objects remains same even after collision. *Momenta is plural of momentum.*

In other words, the sum of momenta of two objects before collision and sum of momenta of two objects after collision are equal.

Mathematical Formulation of Conservation of Momentum:

Suppose that two objects A and B are moving along a straight line in same direction and the velocity of A is greater than the velocity of B.

Let the initial velocity of A=u_{1}

Let the initial velocity of B= u_{2}

Let the mass of A= m1

Let the mass of B=m2

Let both the objects collide after some time and collision lasts for ' t' second.

Let the velocity of A after collision= v_{1}

Let the velocity of B after collision= v_{2}

We know that, Momentum = Mass × Velocity

Therefore,

Momentum of `A(F_A)` before collision `=m_1xxu_1`

Momentum of `B(F_B)` before collision `=m_2xxu_2`

Momentum of A after collision `=m_1xxv_1`

Momentum of B after collision `=m_2xxv_2`

Now, we know that Rate of change of momentum

= Mass × rate of change in velocity

= mass × Change in velocity/time

Therefore, rate of change of momentum of A during collision, `F_(AB)=m_1((v_1-u_1)/t)`

Similarly the rate of change of momentum of B during collision, `F_(BA)=m_2((v_2-u_2)/t)`

Since, according to the Newton's Third Law of Motion, action of the object A (force exerted by A) will be equal to reaction of the object B(force exerted by B). But the force exerted in the course of action and reaction is in opposite direction.

Hence, `F_(AB)=-F_(BA)`

`=>m_1((v_1-u_1)/t)` `=-m_2((v_2-u_2)/t)`

`=>m_1(v_1-u_1)` `=-m_2(v_2-u_2)`

`=>m_1v_1-m_1u_1` `=-m_2v_2+m_2u_2`

`=>m_1v_1+m_2v_2` `=m_1u_1+m_2u_2`

`=>m_1u_1+m_2u_2` `=m_1v_1+m_2v_2` ---(i)

Above equation says that total momentum of object A and B before collision is equal to the total momentum of object A and B after collision. This means there is no loss of momentum, i.e. momentum is conserved. This situation is considered assuming there is no external force acting upon the object.

This is the Law of Conservation of Momentum, which states that in a closed system the total momentum is constant.

In the condition of collision, the velocity of the object which is moving faster is decreased and the velocity of the object which is moving slower is increased after collision. The magnitude of loss of momentum of faster object is equal to the magnitude of gain of momentum by slower object after collision.

## Conservation of Momentum: Practical Applications

- Bullet and Gun: When bullet is fired from a gun, the gun recoils in the opposite direction of bullet. The momentum of bullet is equal to the momentum of gun. Since, the bullet has very small mass compared to the gun, hence velocity of bullet is very high compared to the recoil of the gun. In the case of firing of bullet, law of conservation of momentum is applied as usual.
- In the collision of atoms, the conservation of momentum is applied.
- In the game of snooker, when a ball is hit by stick, the conservation of momentum is applied.
- When the mouth of an inflated balloon is let open, it starts flying, because of conservation of momentum.
- When a cricket ball is hit by bat, the Law of Conservation of Momentum is applied.
- When the coins of carom board are hit by striker, the Law of Conservation of Momentum is applied.
- Newton’s cradle is one of the best examples of conservation of momentum.