# Differential equation for projectile

1. Sep 3, 2014

### Feeh

I want to model a Differential Equation for a projectile motion under 2 forces (gravity and wind)

So, what I have now is an algorithm that simulate the parametric motion (2d) of the project under those 2 forces (given a P position of the projectile with velocity V under a vector of forces F (or acceleration as my projectiles have no mass), find a new position for each time T)
(Link to the motion simulation example: http://s12.postimg.org/wbr8tyej1/projectile.png)

I want to model it as a differential equation. If possible with these variants:

1- basic differential equation projectile motion with no wind, just gravity...the quadratic equation from basic physics should be the solution to it

2- the "complete" form with wind and gravity as I described

3- add drag force. I know that the equation may have no solution

I've learned to solve them but not to model them; if you could give me or point a good text book that show how to model it I'd be glad

Last edited: Sep 3, 2014
2. Sep 3, 2014

### HallsofIvy

Staff Emeritus
So the only force is gravity- since "F= ma" and the force due to gravity is mg, mg= ma, a= g, a constant. The differential equations are $d^2x/dt^2= 0$ since there is no force horizontally and $d^2y/dt^2= -g$ vertically

These two are a lot the same. The force the wind applies is a form of "drag". The one thing to do is to model it as a quadratic function of the speed, $\omega v^2= \omega (v_x^2+ v_y^2)= \omega((dx/dt)^2+ (dy/dt)^2)$. That would give $d^2x/dt^2= -g- \omega ((dx/dt)^2+ (dy/dt)^2)$, $d^2y/dt^2= -g- \omega ((dx/dt)^2+ (dy/dt)^2)$.

3. Sep 3, 2014

### Feeh

Using your example I was able to see how to reach the first example (only gravity) and found the quadratic equation as solution

However I had no time to test the second example (gravity+wind) in fact I did not get the general idea, maybe I'm not seeing something but here is what I've found based on my simulation algorithm and your first example

$$y(t)=y_i + v*t*sin(\beta) - \frac{1}{2}y''(t)*t^2 + w*t^2*sin(\alpha)$$
$$x(t)=x_i + v*t*cos(\beta) + w*t^2*cos(\alpha)$$
position_y = velocity*time - gravity*t^2 + wind*t^2
position_x = velocity*time + wind*t^2

where:
x(t) and y(t) are the projectile position at any given time t
v is the initial velocity of the projectile at launch
β is the launch angle
α is the wind acceleration angle
w is the wind acceleration
xi, yi are the initial positions on a 2d space

This lead me to some other questions:
-Since its me who gives the launch speed (known values), my approach to use them as v and angle β instead of x'(t) and y'(t)? I'm assuming this approach will remove the initial value problem

-The general idea, seems to be correct?

Last edited: Sep 3, 2014