Step-by-step explanation:
Energy is conserved.
Initial KE = final KE + final PE
½ mv₀² = ½ mv² + mgh
v₀² = v² + 2gh
v = √(v₀² − 2gh)
As you can see, the final speed v at height h is independent of the launch angle.
We can also prove this by using kinematics to find the x and y components of the velocity, then using the Pythagorean theorem to find the speed.
In the x direction:
vₓ = v₀ₓ = v₀ cos α
In the y direction:
v² = v₀² + 2aΔy
vᵧ² = (v₀ sin α)² + 2(-g)(h)
vᵧ² = (v₀ sin α)² − 2gh
Pythagorean theorem:
v² = vₓ² + vᵧ²
v² = (v₀ cos α)² + (v₀ sin α)² − 2gh
v² = v₀² cos² α + v₀² sin² α − 2gh
v² = v₀² (cos² α + sin² α) − 2gh
v² = v₀² − 2gh
v = √(v₀² − 2gh)