How can I compute the movement of an object along a parabola?

Question

I am working on a simple 2D game. In the game, I have a 'robot' that throws a ball towards another robot, in the shape of a parabola. Both 'robots' have the same Y coordinate.

The program knows the positions of the two robots, and it knows the position of the parabola's apex ('vertex') as shown here:

I need the ball to travel along the parabola. This means (correct me if I'm wrong), that at any given time, I need to be able to calculate the Y position of the ball, since I know it's X position.

If so, how can I calculate that Y position at any given time, knowing the information above. Please try to make your answers as clear as possible, since my math knowledge is very basic.

Answer 1

So you're looking for a parabolic function y(x) that equals zero at two known points. Let's call those points r₁ and r₂, after robots 1 and 2. One solution is easily found:

y(x) = ( x - r₁ )( x - r₂ )

On top of the robots, one of the terms in parentheses becomes zero, which, multiplied by something else, remains zero. For that reason, we can multiply the entire thing by a factor a. This stretches the parabola vertically (or flips it upside down for negative values) without changing the location of the zeroes. We manipulate a to make sure the curve passes through the vertex, defined by coordinates (x_v, y_v). We solve the following equation for a:

y(x_v) = y_v

a ( x_v - r₁ )( x_v - r₂ ) = y_v

This yields:

a = y_v / (( x_v - r₁ )( x_v - r₂ ))

We get the function y(x) we want by multiplying our original function with the scaling factor a we just obtained:

y(x) = ( x - r₁ )( x - r₂ ) y_v / (( x_v - r₁ )( x_v - r₂ ))

Answer 2

If we are talking about projectiles here, and you know the X position, though it can simply be calculated:

Where d is the horizontal distance from the launch point.

You can simply calculate the y at distance x this equation:

Where:

• x is the x position.
• theta is the angle with the ground.
• g is the gravity constant.
• v is the velocity .