Consider a projectile, launched with initial velocity v0, at an angle θ. To study its motion we may guess that these are the relevant quantities.

(a) Show that {gt/v0, gx/v20, gy/v20, θ} is a complete set of dimensionless products. (One way to go is to find the appropriate free variables in the linear system that arises but there is a shortcut that uses the properties of a basis.)
(b) These two equations of motion for projectiles are familiar: x = v0 cos(θ)t and y = v0 sin(θ)t - (g/2)t2. Manipulate each to rewrite it as a relationship among the dimensionless products of the prior item.

dimensional quantity formula horizontal position x | LiM vertical position y L'MTO initial speed vo L'MT1 angle of launch | Lo MoTo acceleration due to gravity g L'MoT 2 time t oMOT