# Firing a spherical bullet into a watertank

[SOLVED] 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.

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Alfi
Horizontally ?

Or perpendicular to a smooth water surface?

tiny-tim
Homework Helper
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.
Hi TheMan112! Hint: a = dv/dt. You have an equation with separable variables (the form f(y)dy=g(x)dx). This can be solved by integrating both sides: left from y1 to y2, right from x1 to x2.
Another option is solving for v(x):

dv/dx=dv/dt*dt/dx=(dv/dt)/v

After you get v(x) you can get corresponding t from dt=dx/v.

But I don't think that bullets experience linear drag. At high speeds F=-kv^2 is probably a better apoximation.

Last edited:
Horizontally ?

Or perpendicular to a smooth water surface?
Imagine we mounted the muzzle of a gun to a hole in the side of the watertank, fireing into the water horizontally.

Hi TheMan112! Hint: a = dv/dt. Haha, I know that, and from the following differential equation...

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

...I get:

$$\frac{dx}{dt} = e^{\frac{m}{k} t}$$

Which in turn gives:

$$s = \int_{0}^{\infty} v(t) dt = \int_{0}^{\infty} e^{\frac{m}{k} t} dt = -\frac{m}{k}$$

Which gives an answer wrong by a factor of 10^3.

tiny-tim
Homework Helper
$$\frac{dx}{dt} = e^{\frac{m}{k} t}$$

Which in turn gives:

$$s = \int_{0}^{\infty} v(t) dt = \int_{0}^{\infty} e^{\frac{m}{k} t} dt = -\frac{m}{k}$$
Hi TheMan112! Somehow you've managed to get a nearly right answer with the wrong reasoning. It should be:

$$\frac{dx}{dt} = v_0 e^{\frac{-k}{m} t}$$

Which in turn gives:

$$s = \int_{0}^{\infty} v(t) dt = v_0 \int_{0}^{\infty} e^{\frac{-k}{m} t} dt\,=\,...\,?$$ Seems like that did it, thanks!