Calculate launch angle of object moving away from view

The vertical speed at the instance of launch can be calculated by measuring how high the object reaches, that's simply:

height = v^2 sin^2(theta) / 2g

Where v is the initial velocity theta is the angle you're looking for. g is a constant. So that leaves you with three unknowns, and you're going to have to supply two.

You can measure the maximum height through the camera, but the angle of view means that it changes with distance from the camera. Think about a camera aimed at the horizon - if you stand in front of it you take up "this" much of the screen, but as you move away the angle gets less and less.

So to measure the height, you'll need to know the distance of the object into the screen. You might be able to do that if you know the original size of the object.

The horizontal component could be measured using the same size/distance relationship and comparing across several frames. For small trajectories, like a ball across the room, this will be linear enough for reasonable estimates, but it will not work for anything where drag is a real concern.

So if the ball is a known size, take many images of it at difference distances to calibrate your images. Then fire it and find the point where it reaches the maximum altitude. Use that time and the measured distance to calculate the horizontal velocity, the formula above for the vertical, and solve for theta.