# Finding force exerted in an Inelastic Collision

I did a lab today in Physics in which we launched ball from a spring loaded cannon directly into a pendulum that captured the ball, held it, and swung upwards with it (representing a totally inelastic collision). One question in particular has confused me:

If the collision between the projectile and pendulum had lasted 1 millisecond, what would the average force have been which the projectile exerted on the pendulum for the long-range case?

My attempt at a solution is as follows: From all the searching I've been doing online, I've found the equation $F = {{p_f}-{p_i}\over {t}}$. I know $p_f$, $p_i$, and I'm given t. Is my understanding right? Can I go right ahead and crunch these numbers, or do I have an incorrect equation?

-
Hi Mike, and welcome to Physics Stack Exchange! You're close to having a really good question here; the problem is that you just posted the problem but didn't show any attempt to solve or simplify it. As the FAQ says, we don't answer your homework-like questions for you. This is a site for conceptual questions, so what you should do is focus on the particular physical misunderstanding that's keeping you from solving this question yourself, and ask about that. See meta.physics.stackexchange.com/q/714 for more info. Once you fix that, I'll be happy to reopen this. – David Z Apr 16 '12 at 21:45
@David I have made it as general as I think I can, and even found something I think might be the solution. – Mike Gates Apr 16 '12 at 22:14
Great, that's definitely a much better question. – David Z Apr 16 '12 at 22:30
You are right.. – leongz Apr 17 '12 at 1:57
@DavidZaslavsky: Is there any way to answer this without giving the full answer? It's just two formulae--there's no way to "hint" it IMO. Of course, I could explain impulse, but it still amounts to a full solution. – Manishearth Apr 17 '12 at 3:42

Yep, average force $\langle F \rangle=\frac{p_f-p_i}{t}$

There are two things going on here:

## Impulse

For any collision, it is convenient to define a quantity called "impulse". Most collisions consist of large, varying forces acting in a short time. These are hard to calculate, so they can be encoded into the "impulse". The impulse is the change of momentum on a body during a collision. Due to the identity $\vec F=\frac{\rm d\vec p}{\rm dt}$, we get:

$$\vec J=\Delta \vec p=\int\rm{d}\vec p=\int\vec F\rm{d}t$$

Impulses are pretty useful in multi-body problems. Especially when string/friction/etc are present; since we can use them in place of forces and conserve momentum.

## Time-averaging

For any quantity $X$ dependant on time, the time average of the quantity is:

$$\langle X\rangle=\frac{\int X\rm{d}t}{\int \rm{d}t}$$ with limits of integration as the time you want it to be averaged over.

In this case, the numerator becomes the impulse (change in momentum).

Note that for any quantity $X=\frac{\rm{d}y}{\rm{d}t}$, $\langle X\rangle=\frac{\Delta y}{\Delta t}$

This is useful for average speed (total distance traveled by total time), average acceleration, and, of course, average force.

-
Thank you for the exceptional answer! Very helpful and good to know. – Mike Gates Apr 17 '12 at 11:47