Answer:
0.5099646386 radians
Explanation:
Δy = 7.5 m
Δx = 10 m
g = 9.8 m sec⁻²
From trigonometry,
Vy₀ = v₀ sin θ
Vy₀² = v₀² sin²θ
From the equations of constant acceleration kinematics,
Vy₀² = 2g Δy
Two things that are equal to the same thing are equal to each other, so
v₀² sin²θ = 2g Δy
v₀² sin²θ = 147 m² sec⁻²
The double angle formula for the sine,
sin²θ = 0.5 [1 − cos(2θ)]
0.5 v₀² [1 − cos(2θ)] = 147 m² sec⁻²
v₀² = 294 m² sec⁻² / [1 − cos(2θ)]
The range equation,
Δx = v₀² sin(2θ) / (2g)
10 m = 294 m² sec⁻² sin(2θ) / {(2g) [1 − cos(2θ)]}
1.5 sin(2θ) = 1 − cos(2θ)
1.5 sin(2θ) = 1 − √[1−sin²(2θ)]
A convenient substitution.
u = sin(2θ)
2.25 u² = 1 − 2√[1−u²] + 1 − u²
3.25 u² − 2 = −2√[1−u²]
1 − 1.625u² = √[1−u²]
1 − 3.25u² + 2.640625u⁴ = 1−u²
2.640625u⁴ − 2.25u² = 0
As long as u≠0,
2.640625u² − 2.25 = 0
2.640625u² = 2.25
u = 0.8520710059
θ = 0.5 arcsin u
θ = 0.5099646386 radians
Furthermore,
v₀ = 15.16666667 m/s