How can I calculate the future position of my moving object?

Question

I'm relatively new to games development and I have a 2D game where the playing area is an enclosed rectangle with width w and infinite height. In the playing area there is a ball. A gravity force of strength G acts on the ball. The ball starts at position p and has an initial velocity of v.

How would I calculate the position the ball will be at when it reaches the specified height h. I have a function setup using the following parameters:

bool CalculateXPositionAtHeight(float height, Vector2 position, Vector2 velocity, float Gravity, float width, float& xPosition);

• The playing area doesn’t have any boundary at the top, so no collision ever happens with it.
• The ball can be considered as a point.
• There are no precision issues and everything can easily fit in a float.
• There is no air resistance.
• All collisions are perfect without any energy loss.
• If the ball hits the bottom it won’t bounce.

Answer 1

Like Pieter mentioned, you can use 1D kinetic equation for height, which in your case will come to:

\$h = p + vt + 0.5at^2\$

where a is the acceleration due to gravity. You can then use the quadratic equation to solve for t. Once you have t, you can use the kinetic equations again to calculate how far the ball has traveled width ways. This would tell you where the ball was if there were no walls. Assuming the the walls have no friction, it should be pretty obvious how to figure out where the ball would be between the walls.

If the walls do not just directly reflect the velocity, then you need to calculate the time at which the ball hits the wall, and then repeat with the new velocity until the ball reaches the ground.

Assuming gravity is a negative number, the following code should work:

bool CalculateXPositionAtHeight(float height, Vector2 position, Vector2 velocity, float Gravity, float width, float& xPosition)
{
    // kinetic equation for y is: y = position.y + velocity.y * t + 0.5 * Gravity * t^2
    // we want to find when y = height, and have the equation in the form at^2 + bt + c = 0
    // which gives the following:
    float a = Gravity * 0.5f;
    float b = velocity.y;
    float c = position.y - height;
    
    // the quadratic formula can then be used to solve for t
    // we know that gravity will always be negative and time is positive, so only care about that solution
    float t = (sqrtf(b * b - 4.0f * a * c) - b) / (2.0f * a);
    
    // now that we have t, we can solve the kinetic equation for x
    // there is no acceleration in the x direction, so this just becomes:
    float x = position.x + t * velocity.x;
    
    // the following code handles the ball bouncing off the walls
    x = fmodf(x, 2.0f * width);
    if (x > width)
    {
        x = 2.0f * width - x;
    }
    
    xPosition = x;
    
    return true;
}

Answer 2

@Richard Byron explained well and gave a well-made sample code.

I think it's better to add additional information about the basic math.
I leaved out the wall collision.

Sorry for bad hand writing an drawing.

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