Chapter 8.7, Problem 78E

Projectile motion. Eliminate the parameter $t$ from me parametric equations $x=\left(v_0\cos \theta \right)t,y=\left(v_0\sin \theta \right)t-\frac{1}{2}g{t}^2$ to obtain a Cartesian equation.

a. Show that the trajectory of a projectile is a parabola.

b. Find the vertex of the parabola.

c. Find the line or flight of the projectile.

d. Find the range.

e. Show that for a fixed Initial speed $v_0$, the maximum range is obtained when $\theta =45°$. Find a formula for this maximum range.

f. Show that doubling the initial speed of a projectile has the effect of multiplying both the range and the maximum height by a factor of four.