# Projectile motion model

In my studies regarding projectile motion, I recently completed a simple level question surrounding a multileveled golf range. I had to determine the trajectories of these balls after they were shot into the air.

I was wondering if some sort of mathematical model could be developed for finding the horizontal distance, d in terms of v (initial velocity), h (initial height) and θ (angle to the horizontal) from each of the three levels, if the bottom level was treated as 0, the middle level, h, and the top level 2h.

Thankyou

• Welcome to MSE. What have you tried? Can you at least update your question with the equations of the projectile? – mathcounterexamples.net May 2 at 8:55

## 2 Answers

If a projectile starts out at position $$(0,h)$$, initial velocity $$v$$ and angle to the horizontal $$\theta$$ then at time $$t$$ it has co-ordinates $$(x(t),y(t))$$ where

$$x(t) = vt \cos \theta \\ y(t) = h + vt \sin \theta - \frac 1 2 gt^2$$

We can eliminate $$t$$ and write the equation of the projectile's path as

$$y = h + x \tan \theta - \frac{g}{2v^2 \cos^2 \theta}x^2$$

Set $$y=0$$ and you are left with a quadratic equation in $$x$$. Solve for $$x$$ and you have the two values of $$x$$ when the projectile is at ground level. Only one of these values is positive, so that is the one you want.

A good way of thinking about this is thinking about the time the ball is in the air.

We know that $$v_x = v\cos\theta$$, so multiplying this by the total 'air time' will give the range of the projectile.

For our hieght, we have $$y=-\frac{g}{2}t^2 + vt\sin\theta+y_0$$, where $$y_0$$ is the initial height.

To get the total flight time, we simply substitute $$y=0$$ (where the projectile will hit the ground), and solve for $$t$$ using quadratic formula. Then multiplying by $$v_x$$ as mentioned before will give the range.