Two Guys Arguing

mario math

Posted in java, mario, math by youngnh on 08.13.09

To calculate Mario’s velocity, with KEY_RIGHT pressed down continuously for n frames:

let (1.2 if KEY_SPEED is held down as well)

at frame one, then, your velocity

frame 2:
frame 3:
frame 4:

which, if you multiply through, looks like this:

Which makes it a bit easier to generalize Mario’s velocity at any frame n:

we know from paying attention in high school wikipedia that:

however, our sequence only goes to n-1 frames. It’s missing the last term:

subtracting xn from both sides

which we could have maybe figured out faster if we had remembered this identity:

replacing the lower m with 0 and upper n with n-1 we get:

which multiplied by is what we found earlier.

and so mario’s velocity after n frames of acceleration is:

which can be calculated in constant time by this Java function:

public float velocityAfter(float accel, int nFrames) {
return accel * ((1 - Math.pow(0.89, nFrames)) / 0.11);
}


Whew.

Thanks to Heath Borders and my co-blogger Ben for working this out on a whiteboard with me.