What is the vertical component of the projectile’s initial velocity?

Question

The straight-line distance d and angle φ with respect to the horizontal are known for a target. A projectile is fired at some unknown angle θ such that it reaches the target at the top of its parabolic path. Known quantities: φ, d, and g .

Answer 1

The height of the target is d sinφ. This quantity must equal the maximum height of the trajectory parabola, since the projectile reaches the target at the top of the parabola. From kinematics, the maximum height of the trajectory is

H = v₀²sin²θ / (2g)

d sinφ = v₀²sin²θ / (2g)

The vertical component of the initial velocity is v₀sinθ.

d sinφ = (v₀sinθ)² / (2g)

v₀sinθ = √(2gd sinφ)