Last updated on November 21st, 2021 at 10:13 am

**In this post, we will see how Newton’s third law can be derived from the equation of conservation of momentum.**

## Derive Newton’s third law from the equation of conservation of linear momentum

Say, a body of mass m1 with velocity u1 collides with another body of mass m2 with velocity u2. After the collision, say, mass m1 moves with velocity v1, and mass m2 moves with v2 velocity.

As per the principle of conservation of momentum, we can write the following equation:

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

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

=> – m_{1} ( v_{1} – u_{1} ) = m_{2}(v_{2} – u_{2})

=> – m_{1} ( v_{1} – u_{1} )/Δ = m_{2}(v_{2} – u_{2})/Δ

**=> – (Rate of change of momentum of mass m1) = (Rate of change of momentum of mass m2) **

=> – F_{1} = F_{2}

This is the expression of Newton’s third law.

Thus Newton’s third law can be derived from the principle of conservation of linear momentum.

## Related study (collisions)

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

**Numerical problems – collisions**

**Numerical problems – 2D collisions**