## mario math

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 *x*^{n} 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.

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