Jump-Arc Designer | Explorations

The whole jump is two lines, run once per frame: vy += 0.5, then y += vy. Step through it and watch both numbers.

Jump strength

Frame

0

vy this frame
0

y moved by

0

y is now

300

Standing on the ground at y = 300. Press Step to jump.

frame	vy	moved	y

Apex height
—

Frames going up

—

Target

—

The number

Jump strength 12

Gravity 0.5

Fixed. Every number in the lesson is built on it, and the rule only works because of it.

The target

Target height 144 px

Your jump log

strength	height	frames up

Look at your log. When the strength doubles, what happens to the height? Then work out strength × (strength minus a half) for each row and compare.

A jump is one shove, gravity is two adds

Every frame, the game does exactly this:

vy = vy + gravity then y = y + vy

Gravity adds to your speed. Your speed adds to your position. A jump is a one-time kick: jump(player, 12) sets vy = -12, and up is negative. That's the whole engine. Everything else is a consequence.

How long do you rise?

You leave the ground with speed J going up. Gravity eats g of that speed every frame. So the speed runs out after J ÷ g frames. With our course gravity of 0.5, a jump of strength 12 rises for 12 ÷ 0.5 = 24 frames.

How HIGH do you get?

Take a jump of 12 apart. The velocity starts at 12 upward and gravity takes half off it every frame, so the distances the player moves on the way up are 11.5, 11, 10.5, and so on down to 0.5 and 0. Twenty four numbers, and their total is the height. Do not add them in order: pair the ends. 11.5 with 0, 11 with 0.5, 10.5 with 1. Every pair makes 11.5, and 24 numbers is 12 pairs, so the height is 12 × 11.5 = 138. Exactly, not about. The Fold the Steps tab draws it.

Do that with no number picked and, at a gravity of 0.5, it collapses to one rule: height = strength × (strength minus a half). Check it: 4 × 3.5 = 14, 8 × 7.5 = 60, 16 × 15.5 = 248, the same numbers the Jump Lab measures.

Strength shows up twice in that rule, which is why doubling the strength makes the jump about four times higher, and why the arc is the shape math class calls a parabola.

Going backwards

You usually have a height in mind and want the strength. Ignore the half and undo the squaring: strength ≈ √height. Want 144? √144 = 12. But 12 reaches 12 × 11.5 = 138, six short, because the square root always drops that half a strength. So round up: 13 reaches 13 × 12.5 = 162.5 and clears the line. That is not a fudge, it is the exact rule telling you the square root comes up short by half a strength, every time.

Where you'll use this

In your engine it is one line, and it works only because your gravity is 0.5:

function jumpFor(height) { return Math.sqrt(height); }

let JUMP = jumpFor(TARGET) + 1;

Next week the platforms go in the air, and the first question about each one is whether the player can reach it. That is this rule run backwards, and you will ask it every time you place a ledge.