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

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

Range (R): Horizontal distance at t = T: R = ucos(theta) * T = ucos(theta) * 2usin(theta)/g R = u^2 * sin(2theta)/g

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