Trajectory Position At A Given Time

[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

 

[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.

 

[Integral]

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
$$v_y(0) = v_i(0) sin (\theta)$$

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

 

[Janitor]

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