# Confusion with force and time

It is said that force is an interaction that causes an object to accelerate. It is measured in newtons. $$1$$ N of force causes an object with a mass of $$1$$ kg to accelerate at $$1$$ m/s². Now let's imagine a box sliding on frictionless surface and hitting a standing box. Standing box will accelerate. But their interaction will last extremely short amount of time. Isn't the length of time of them touching each other approaching zero? If it is, isn't the acceleration approaching zero?

Intuitively we know it will move, but mathematically it confuses me.

This is where it's better to use $$\Delta p = F\Delta t$$, often refered to as "impulse". If the second box weighs 1kg and accelerates to 10 m/s, that's a change in momentum of $$\Delta p = 10 kg \cdot m/s$$. If you know they were in contact for only 0.01 seconds, then the force must be $$F = \frac{\Delta p}{\Delta t} = {10}{0.01} = 1000N.$$
Realistically the two boxes would be in contact longer than that - maybe 0.1-0.3 or so, resulting in a much lower force. However if the boxes are made out of very hard incompressible material - say blocks of diamond - then the $$\Delta t$$ would be very small and the force very high.
This is why it hurts more to run into a brick wall ($$\Delta t$$ small) than a mattress ($$\Delta t$$ large) even though the total integrated acceleration ends up being the same.