Step-by-step explanation:
Given:
x₀ = 0 m
x = 23 m
y₀ = 0 m
y = 16 m
v₀ᵧ = v₀ₓ tan 60° = v₀ₓ √3
aₓ = 0 m/s²
aᵧ = -9.8 m/s²
Find: vₓ, vᵧ
First, in the x direction:
Δx = v₀ₓ t + ½ aₓ t²
23 = v₀ₓ t + ½ (0) t²
t = 23 / v₀ₓ
Next, in the y direction:
Δy = v₀ᵧ t + ½ aᵧ t²
16 = v₀ᵧ t + ½ (-9.8) t²
16 = v₀ᵧ t − 4.9 t²
Substitute:
16 = (v₀ₓ √3) (23 / v₀ₓ) − 4.9 (23 / v₀ₓ)²
16 = 23√3 − 2592.1 / v₀ₓ²
2592.1 / v₀ₓ² = 23√3 − 16
v₀ₓ² = 2592.1 / (23√3 − 16)
v₀ₓ = 10.43 m/s
Now find v₀ᵧ and t:
v₀ᵧ = v₀ₓ √3
v₀ᵧ = 18.06 m/s
t = 23 / v₀ₓ
t = 2.206 s
Use these to find the final velocities.
vₓ = aₓ t + v₀ₓ
vₓ = (0) (2.206) + 10.43
vₓ = 10.43 m/s
vᵧ = aᵧ t + v₀ᵧ
vᵧ = (-9.8) (2.206) + 18.06
vᵧ = -3.55 m/s
Use Pythagorean theorem to find the magnitude:
v² = vₓ² + vᵧ²
v² = (10.43)² + (-3.55)²
v = 11.02 m/s
Use trig to find the direction:
tan θ = vᵧ / vₓ
tan θ = -3.55 / 10.43
θ = -18.82°