Time of Flight (T): When projectile returns to launch height, y = 0. 0 = usin(theta)T - $\frac{1}{2}$g$T^2$ T(usin(theta) - gT/2) = 0 T = 0 (launch) or T = 2u*sin$\frac{theta}{g}$

Maximum Height (H): When $v_y$ = 0: 0 = usin(theta) - g$t_H$ => $t_H$ = usin$\frac{theta}{g}$ = T/2 H = usin(theta) * $t_H$ - $\frac{1}{2}$g*$t_H^2$ H = $u^2$*sin^2$$\frac{theta}{(2g)}

Range (R): Horizontal distance at t = T: R = ucos(theta) * T = ucos(theta) * 2u*sin$\frac{theta}{g}$ R = $u^2$ * sin$\frac{2*theta}{g}$

Key insight: R is maximum when sin(2*theta) = 1, i.e., theta = 45 degrees.