Calculating velocity to reach the jump height while the player presses the jump button

Question

The height of the jump depends on how long the player presses the jump button. There are only two parameters - height and time. Gravity is constant.

Let's say that the height has three units, and the time is set to a second.

There are fixed time-segments during which velocity must be added. If the player presses the button for the entire second, they will reach the three units height, taking into account the fact that gravity is pulling them down.

When I tried to solve this problem, I was trying to use v = sqrt(2gH) formula which was perfect to calculate instant velocity for fixed-height jumps.

Not for this case! My code may look like this:

private float jumpHeight = 3f;
private float jumpTime = 1f;

private const float GRAVITY = -10f;

...

private float timer = jumpTime;

public void FixedUpdate(float fixedDeltaTime) {
    if(timer < 0) {
        return;
    }
    timer -= fixedDeltaTime;
    
    var deltaTimeNormalized = fixedDeltaTime / jumpTime;
    var deltaHeight = jumpHeight * deltaTimeNormalized;
    var jumpVelocityY = CalculateJumpSpeed(deltaHeight, Math.Abs(GRAVITY));

    character.rigidbody2D.velocity.y += jumpVelocityY;
}

private float CalculateJumpSpeed(float height, float gravity) {
    return Math.Sqrt(2 * height * gravity);
}

The controlled character skyrockets, and I don't quite understand the reason for this behavior.

I clearly see that the speed is gaining rapidly, but I cannot understand why the original idea does not work, and what other solutions there might be that would take the same two parameters.

Answer 1

Your math is not mathing.

sqrt() is not a linear operation. This is what the slope of a sqrt(x) looks like:

This is what a linear operation would look like

If I want to jump up 1 unit of height with a gravity of 10 units/s then the velocity I Need is sqrt(2 * 10 * 1) = 4.47214... If I want to jump up 3 units of height then the velocity I need is sqrt(2 * 10 * 3) or 7.74597... Which is not 3 * 4.47214.

Answer 2

You're adding to the speed in every FixedUpdate with this line. character.rigidbody2D.velocity.y += jumpVelocityY;

What I'd do is treat it like a regular instant velocity, fixed-height jump. Then, if the jump button is released, gravity would increase to end the jump earlier.