Consider the following variables for a projectile:
- launch velocity $v$
- launch angle $\theta$
- horizontal range $r$
- maximum height $h$
- total time taken $t$
It can be easily shown that the projectile can be defined by specifying any two of the variables, i.e. given any two of the variables, the other three can be expressed in terms of the given two.
The only exception is the case where $h, t$ are specified.
Questions:
Is the conclusion above correct, and why so? What is the relationship between $v,\theta, r$ for given values of $h,t$?
Note that $h=\frac 18 gt^2$, hence either one can be expressed in terms of the other, whereas the other variables have to be expressed in terms of two other given variables.
NB: $g$ is considered a constant in the above, and not a variable.
Two different trajectories with the same $v$ and $r$ would be a counter example. So, for launch angles of $30$ degrees and $60$ degrees we get the same range $r$ for the same launch velocity $v$.