Why doesn't this 2d physics calculation yield the desired jump height?

Question

I found a formula for gravity and jumping in this article: http://error454.com/2013/10/23/platformer-physics-101-and-the-3-fundamental-equations-of-platformers/

I implemented this formula and proceeded to do some test calculations, but they yield incorrect results. I expected the outcome to be 4, but the test resulted in 3.8. What am I doing wrong? How do I get the number 4?

float deltaTime = 1f / 50f;

float maxJumpHeight = 4;
float timeToApex = 0.44f;
float gravity = (2 * maxJumpHeight) / (timeToApex * timeToApex);

float jumpHeight = 4;
float jumpVelocity = (float)Math.Sqrt(2 * gravity * jumpHeight);

float velocity = -jumpVelocity;
float position = 0;

while (velocity < 0)
{
    velocity += gravity * deltaTime;
    position += velocity * deltaTime;
    Console.WriteLine("Position = {0}", position);
}

Answer 1

It's because of the discreet time and the way you integrate. Because you step time forward at 1/50 of a second you're not guaranteed to hit the actual apex of 4.0, the delta is just not granular enough. (Try setting the delta time to 1/5000 and you'll see that it gets closer to 4, obviously that won't work for your game but it will show you the effect).

Depending on the integration method you use you will get varying degrees of correctness, with Euler integration being quite bad when it comes to accuracy.

This article has a nice explanation on why integration might lose precision; http://www.marekfiser.com/Projects/Real-time-visualization-of-3D-vector-field-with-CUDA/4-Vector-field-integrators-for-stream-line-visualization

Answer 2

I was able to achieve a precise gravity simulation by applying the 'correction' said in this article: http://www.niksula.hut.fi/~hkankaan/Homepages/gravity.html

Basicaly it consists in dividing the gravity by 2 in the very first frame/iteration when adding it to velocity.