# 3.2.2 Average Force (Fave = Δp/Δt)

Have you ever been hit by a hard object like baseball or golf ball? It hurts more if it hits your head rather than your butt, right? Have you ever wondered why?

Consider a 2 kg ball colliding into a brick wall at 3 m s-1, and rebounding at 3 m s-1 after the collision.

Obviously, there is a contact force between the ball and wall during the collision. Assuming the duration of collision to be $\Delta t=0.2\text{ s}$, we can calculate the (average) value of the impact force experienced by the ball using

$\displaystyle {{F}_{{ave}}}=\frac{{\Delta p}}{{\Delta t}}=\frac{{m\Delta v}}{{\Delta t}}=\frac{{m({{v}_{f}}-{{v}_{i}})}}{{\Delta t}}=\frac{{2(-3-3)}}{{0.2}}=-60\text{ N}$

The negative sign indicates that the wall exerted a leftward 60 N force on the ball. By N3L, the ball must have exerted a rightward force of 60 N on the wall.

Now what if the ball hits a cardboard instead? This time, the ball would probably not rebound. Let’s assume that the ball simply comes to rest and sticks on the cardboard after $\Delta t=0.4\text{ s}$.

$\displaystyle {{F}_{{ave}}}=\frac{{\Delta p}}{{\Delta t}}=\frac{{m\Delta v}}{{\Delta t}}=\frac{{m({{v}_{f}}-{{v}_{i}})}}{{\Delta t}}=\frac{{2(0-3)}}{{0.4}}=-15\text{ N}$

So during the impact, the ball and the wall exerted an average force of 15 N on each other.

So you have only your rigid skull to blame for causing so much ∆p in such short ∆t to the ball, causing both you and the ball to suffer a large and painful impact force. On the other hand, your fleshy bum takes its own sweet and longer ∆t to cause a smaller ∆p to the ball, which translates to a smaller impact force between you and the ball.

Demonstration

Eggs, Wall and Bedsheet

Concept Test

0437