What is the maths behind projectiles (including air resistance)?

1. Oct 10, 2008

Georgepowell

E.g. A football of mass 1kg is thrown with a speed of 10m/s at an angle of 30 degrees from a horizontal plane. Take air resistance (in newtons) as 1/15 th of the speed of the ball in the exact opposite direction that the ball is traveling. How long is the ball in the air? and how far (horizontally) has it traveled at the point of landing?

I understand how to figure it out if I can ignore air resistance, but how would I answer this question?

I am not so much interested in the answer, but more how to figure out the answer, I have just given you that example so it gives you something to answer if you need an example.

Last edited: Oct 10, 2008
2. Oct 10, 2008

atyy

3. Jan 25, 2009

Georgepowell

That tells me how to figure out the air resistance, and not how to predict its movement after I know it.

Anybody else?

4. Jan 25, 2009

Hootenanny

Staff Emeritus
One would need to formulate a system of differential equations using Newton's second law. The exercise is fairly straight forward for the case of linear drag (as you have described in your opening post).

5. Jan 25, 2009

Georgepowell

Ok, so from what I already know I can write the instantaneous acceleration in terms of the velocity at that time, but I can't seem to figure out how to have acceleration/velocity as a function of time.

Any help on this?

6. Jan 25, 2009

Hootenanny

Staff Emeritus
We have Newton's second law:

$$\mathbf{F} = m\frac{d^2\mathbf{r}}{dt^2}$$

Where r = xi + yj is position, where x=x(t) and y=y(t). Splitting into vertical and horizontal components:

$$-mg -\kappa y^\prime = my^{\prime\prime}$$

$$-\kappa x^\prime = mx^{\prime\prime}$$

Where $\kappa = 1/15$. In canonical form:

$$y^{\prime\prime} + \frac{\kappa}{m} y^\prime = g$$

$$x^{\prime\prime} + \frac{\kappa}{m} x^\prime = 0$$

Both of the above are second order differential equations with constant coefficients. Both are separable and straightforward to solve.

7. Apr 1, 2009