# Firing a spherical bullet into a watertank

## Main Question or Discussion Point

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.

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Andy Resnick
The differential equation to solve is:

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

And solutions are fairly straightforward to find.

malawi_glenn
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