# Trajectory Position At A Given Time

1. Mar 28, 2004

### sklar

I am writing a graphics program to model projectile motion of a ball being thrown. What I am looking for is an equation that will give a specific X and Y coordinate, in the parabolic path, for any specific time it is given, and a arbitratry velocity and angle.

For example, if I give it some arbitrary velocity and angle, I would like to be able to figure out how high in the air the ball is and how far it has traveled horizontally at 5 seconds, 13.4 seconds or any other random time.

Any help would be appreciated.

Sklar

2. Mar 28, 2004

### Janitor

What sort of simplifying assumptions are you making? A flat earth, constant acceleration of gravity w.r.t. altitude, no air drag? Since you mention "parabolic path," I imagine you are going with all of these idealizations.

If you idealize (simplify) things enough, you can make use of closed-form solutions that were worked out with calculus centuries ago. If you are going to make it more realistic, you will probably have to resort to numerical approximation methods.

3. Mar 28, 2004

### Integral

Staff Emeritus
A strictly time depentent solution is pretty easy. The only force acting is gravity. Application of Newtons laws gives.

in the x direction

$$v_x (0) =v_i(0) cos(\theta)$$
so
$$x(t) = v_x(0)t + C$$

If you define the origion as the point where the ball is thrown C=0

Gravity acts in the y direction so

$$\frac {d^2y} {dt^2} = -g$$

$$\frac {dy} {dt} = -gt + v_y(0)$$
but
$$v_y(0) = v_i(0) sin (\theta)$$

$$y(t) = Y_0 + v_y(0)t - \frac {g t^2} 2$$

Last edited: Mar 28, 2004
4. Mar 28, 2004

### Janitor

And thereby Integral has derived your parabolic curve, parameterized by time t.