# Collisions and Newton’s Laws of Motion – How to relate these?

Last updated on November 21st, 2021 at 10:07 am

This post covers how Collisions and Newton’s Laws of Motion are related. As collisions, we can take examples of the interaction between two bodies (gun-bullet, skater-skateboard, hose-water).

For simplicity let us consider a simple collision between two balls and find out how Newton’s Laws of Motion can be applied to analyze the events related to a collision.

**Collisions and Newton’s Laws of Motion**

Let us apply Newton’s three laws to this problem.

**Newton’s first law** is applied during a collision – how?

As per the diagram above, the red ball of mass m_{1} is moving towards the right with u_{1} velocity, and the blue ball of mass m_{2} is also moving towards the right but with a velocity u_{2} .

In the collision, the red ball slows down to v_{1} and the blue ball speeds up to v_{2}. Newton’s first law tells us that this means there is a force acting on the red ball (F_{1}) directing towards the left (against its motion) and thus the red ball slows down.

Similarly, a force is acting on the blue ball (F_{2}) directing towards the right (along its initial direction) and helps the blue ball to speed up.

This is what is said in Newton’s first law of motion, that a force is required to change the velocity of a motion.

**Newton’s second law** is applied during a collision – how?

This law tells us that the force will be equal to the rate of change of momentum of the balls involved. now if the balls are touching each other for a time ∆t.

So we can write, F_{1} = (m_{1}v_{1} -m_{1}u_{1}) / ∆t ……… (1)

and, F_{2} = (m_{2}v_{2} -m_{2}u_{2}) / ∆t ……… (2)

**Newton’s third law** is applied during a collision – how?

According to the third law, if the red ball exerts a force on the blue ball, then the blue ball will exert an equal and opposite force on the red ball.

**That means, F _{1} = – F_{2}**

From equation (1) and (2) we can write,

(m_{1}v_{1} -m_{1}u_{1}) / ∆t = – (m_{2}v_{2} -m_{2}u_{2}) / ∆t

=> m_{1}u_{1} + m_{2}u_{2} = m_{1}v_{1} + m_{2}v_{2}

**In other words, the total momentum before the collision = total momentum after the collision.**

This is what is stated by the Law of conservation of linear momentum.

## Related study (collisions)

**Collisions – definitions, types, sample numerical**

**Numerical problems – collisions**

**Numerical problems – 2D collisions**