Drop an egg on a pillow and on a hard floor from the same height. It hits both at the same speed and stops in both cases, so its change in momentum is the same. Why does only one break? The answer combines momentum with the second law.
Key idea
Force × time = change in momentum: $$F\,t = \Delta p \quad\text{so}\quad F = \dfrac{\Delta p}{t}$$ For the same change in momentum, a longer stopping time means a smaller force.
Why it works
From $F = ma$ and $a = \dfrac{\Delta v}{t}$, we get $F = \dfrac{m\,\Delta v}{t} = \dfrac{\Delta p}{t}$. The floor stops the egg in a tiny fraction of a second, so the force is huge. The pillow sinks in and takes many times longer, so the force is many times smaller.
Worked example: A hard stop
A 0.5 kg ball moving at 10 m/s is stopped by a wall in 0.01 s. Find the average force.
- Change in momentum: $\Delta p = 0.5 \times 10 = 5$ kg·m/s.
- $F = \dfrac{5}{0.01} = 500$ N.
Answer: 500 N
Worked example: A soft stop
The same ball is caught by a padded mitt that takes 0.1 s to stop it. Find the force.
- Same $\Delta p = 5$ kg·m/s.
- $F = \dfrac{5}{0.1} = 50$ N: ten times the time, one tenth the force.
Answer: 50 N
Common mistake
Thinking padding reduces the change in momentum. The object must go from moving to stopped either way, so $\Delta p$ is the same. Padding, crumple zones and airbags spread that same change over more time, which lowers the force. Check: the speed before and after didn't change, so neither did $\Delta p$.
This is why a car's front end is built to crumple and why an airbag inflates: both stretch the stop over more time. A gymnast's mat and a catcher who pulls the glove back work the same way.
Drop an egg on a pillow and a floor in the Science Lab: Momentum and Collisions.