Discrete-time projectile motion calculator

This is a tool wich some friends and I worked on. Its objective is to provide formulas for projectile motion, but in discrete time. We like to call the theory behind the calculator BiLoMa’s theorem. The name of the theorem originates from our nicknames, BisUmTo, Lore, and Mamo (me!).

We initially worked on this theorem because of Minecraft, to calculate the initial velocity needed to launch blocks of sand in such a way that they land at an exact point in space. If you own Minecraft and have some knowledge of command blocks, you can fill out the form below and see how our theorem works in practice.

Required initial velocity:
Minecraft command:

BiLoMa’s theorem

Let n be the number of time units that have passed since the object was launched and a the (gravitational) acceleration. Assuming that the object was launched with an initial position of 0 and an initial velocity of v0, we have the following formulas for the velocity vn and the position pn:

BiLoMa’s theorem

The damping factor d is used in Minecraft to prevent objects from incrementing their velocity indefinitely. Using these formulas with d → 1 means having no damping at all.

Proof of the formula for vn

Proof of the formula for vn

Proof of the formula for pn

Proof of the formula for pn