Exercise 1

Consider a projectile, launched with initial velocity $v_{0}$, at an angle $\theta$. To study its motion we may guess that these are the relevant quantities.

• Show that $\{\hskip 1.00006ptgt/v_{0},gx/v_{0}^{2},gy/v_{0}^{2},\theta\hskip 1.00006pt\}$ is a complete set of dimensionless products. (Hint. 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.)

• These two equations of motion for projectiles are familiar: $x=v_{0}\cos(\theta)t$ and $y=v_{0}\sin(\theta)t-(g/2)t^{2}$. Manipulate each to rewrite it as a relationship among the dimensionless products of the prior item.