What is the Law of Conservation of Linear Momentum?

The law of conservation of linear momentum states that the total momentum of two or more bodies in an isolated system would be conserved unless an external force is applied. It is crucial to note that it is true in the case of an “isolated system” where no force is acting on the objects. The other interpretation of this law is that when two objects coming from opposite directions collide, their total initial linear momentum before the collision is equal to the total final momentum after the collision. The law of conservation of momentum is closely related to Newton’s third law of motion, which states that there is an equal and opposite reaction to every action.

The Law of Conservation of Momentum and Newton’s Third Law

Newton’s third law states that whenever a force is applied to an object, an equal and opposing force is applied by that object. For example, if a hammer exerts a force on a brick, then that brick would exert an opposing force of equal magnitude on that hammer. 

In mathematical terms, it can be written as FAB = -FBA (here, FAB is the force applied by A on B and FBA is the force applied by B on A). The negative sign indicates that the forces are opposite in direction.

Now, the length of time when the forces act on the two objects is the same. Let’s say that the force applied by object B on A is for 1 millisecond. Then, A’s force on B would also be for 1 millisecond only. 

Thus, the impulse felt by objects A and B would be equal in magnitude.

FAB tAB = -FBA tBA 

Impulse is defined as the change in momentum of the object. So, the two objects experience an equal amount of change in their respective momentums (since their impulse is the same). 

Change in Momentum, △p = m△v (mass x change in velocity)

Since, △pA = -△pB 

Or mA△vA = -mB△vB 

Derivation of Conservation of Linear Momentum

Let there be two objects, A and B, of mass mA and mB, respectively. Now, let the initial velocity of A be A1, and the final velocity of A be A2. Let the initial velocity of B be B1, and the final velocity of B be B2. 

Let the two objects collide with each other as they travel in opposite directions.

Change in the momentum of A, △pA = mA (A2 – A1)

Change in the momentum of B, △pB = mB (B2 – B1)

Since the momentum is conserved, △pA = – △pB 

mA (A2 – A1) = -mB (B2 – B1)

This equation can be rearranged to give mAA2 + mBB2 = mBB1 +mAA1

Collision of two objects

The law of conservation of momentum is especially relevant when talking about the event of two objects colliding. The objects can be of either varying or identical masses and of varying or the same velocity. In any case, their total initial linear momentum is equal to their total final momentum.

When two objects of the same mass and velocity collide with each other, then there is no change in their velocity after the collision.

When two objects of differing mass but the same velocity collide, then the velocity of the heavier object decreases after the collision as some of the momentum is transferred to the lighter body. There is a transfer of momentum taking place, but the overall momentum of the system remains the same. 

When two objects collide such that one of them is at rest and the other one is approaching it, then there is a transfer of momentum such that either both of them move in one direction or one of them comes to rest while the other one starts moving. 

Applications

1. The recoil of a gun: When a bullet is fired from the gun, the gun recoils due to the conservation of linear momentum. This is why extreme caution has to be taken while firing a gun, as it can harm the shooter while recoiling.

2. Propulsion of a rocket: A rocket goes up high in the sky because a large amount of gas is ejected from its bottom end. In order to conserve the momentum, the rocket moves up in the opposite direction and hence, flies up.

Conclusion

The law of conservation of momentum is an important law in physics that helps explain several phenomena. This law is especially crucial when we talk about the collision of two objects. Usually, the collision of two objects can be considered close to an isolated system because the time duration for collision is so small that the impact of external forces like friction is almost negligible.