# Firing a spherical bullet into a watertank

I've got a problem, involving non-constant acceleration:

If we fire a spherical bullet horizontally into a watertank, how far will the bullet traverse?

I've figured as much that a spherical bullet provides a retarding force:

$$F = -k \cdot v$$ where k is a constant.

This should provide the following non-constant acceleration due to Newtons 2nd law.

$$a = \frac{F}{m} = - {\frac{k v}{m}}$$

I'm thinking I should integrate two times over a(t) to get an expression for x(t), but since "a" is proportional to v(t) and not directly to t, I dont know how to do it without getting a recursive expression.

## Answers and Replies

Andy Resnick
Science Advisor
Education Advisor
The differential equation to solve is:

$$m \ddot{x} - k \dot{x} = 0$$

And solutions are fairly straightforward to find.

malawi_glenn
Science Advisor
Homework Helper
you already posted this in HW-section.

Yes, my apologies. I started this thread before I noticed one should ask such questions i the HW-section. I'm going to post my reply to Andy there, you may remove this thread.

malawi_glenn
Science Advisor
Homework Helper
Yes, my apologies. I started this thread before I noticed one should ask such questions i the HW-section. I'm going to post my reply to Andy there, you may remove this thread.

It's ok, threads are often moved to the correct place after awhile. Just wanted to draw your attention to this :-)