# Is Newton's third law due to inertia?

Newton's third law states that each force has an equal and opposite force. If I kicked a ball, it would apply the same force on me. Is this due to the ball's inertia? To clarify, is the ball exerting a force on me because it wants to stay in its original position?

• – leongz Apr 27 '18 at 21:39

Newton's third law is a consequence of conservation of momentum, which we have never been able to falsify. Consider a system of two particles with total momentum $\vec P$ such that

$$\vec P = \vec p_1 + \vec p_2.$$

Noting the relationship between force and impulse, $\vec F = \frac{d\vec p}{dt}$, we can take a time derivative to find

$$\frac{d\vec P}{dt} = \frac{d\vec p_1}{dt} + \frac{d\vec p_2}{dt} = \vec F_{\rm 2\, on\, 1} + \vec F_{\rm 1\, on\, 2}.$$

If the total momentum is conserved, then $\frac{d \vec P}{dt} = 0$, and we have $$\vec F_{\rm 2\, on\, 1} = - \vec F_{\rm 1\, on\, 2}.$$

The ball exerts a force back on you in order to conserve linear momentum.

• What about a static case? – V.F. Apr 28 '18 at 1:58
• V.F. The total momentum is still constant. – Bill N Apr 28 '18 at 2:31
• I don't disagree with that, but in your derivation, you rely on the relationship F=dp/dt for each particle, which does not seem to be valid in the static case. – V.F. Apr 28 '18 at 14:41