Way back in 2016, I gave a talk at GDC titled ‘Math for Game Programmers: Building a Better Jump.’ In the years since, I’ve solved some of those equations again, but I had never written down the entire table of formulas until just recently. Given the following symbols, this is every solution for any one metric in terms of any other two.

• Initial velocity: v₀
• Gravity: g
• Apex height: a
• Time to apex: t_a

To solve for…	…in terms of…	…use this formula:
v0	g, a	v0 = √(-2ga)
v0	g, ta	v0 = –gta
v0	a, ta	v0 = 2a / ta
g	v0, a	g = –v02 / 2a
g	v0, ta	g = –v0 / ta
g	a, ta	g = -2a / ta2
a	v0, g	a = –v02 / 2g
a	v0, ta	a = v0ta / 2
a	g, ta	a = –gta2 / 2
ta	v0, g	ta = –v0 / g
ta	v0, a	ta = 2a / v0
ta	g, a	ta = 2a / √(-2ga)

We can also introduce metrics for lateral movement.

• Lateral distance to apex: x_a
• Lateral speed: v_x

In all cases, time to apex t_a may be substituted for x_a / v_x.

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