Question

You are firing at a target on the ground 500m away. You are on a 50m high stand. Your barrel is fixed at 20 degrees to the horizontal. What should the initial velocity of the bullet be if it is to hit the target? Assume gravity is 10 m/s2 in your working out for simplicity.

Answer 1

80 m/s

Step-by-step explanation:

Given:

x₀ = 0 m

x = 500 m

y₀ = 50 m

y = 0 m

v₀ₓ = v₀ cos 20°

v₀ᵧ = v₀ sin 20°

aₓ = 0 m/s²

aᵧ = -10 m/s²

Find: v₀

In the y direction:

y = y₀ + v₀ᵧ t + ½ aᵧ t²

0 = 50 + (v₀ sin 20°) t + ½ (-10) t²

0 = 50 + v₀ sin 20° t − 5t²

5t² − 50 = v₀ sin 20° t

In the x direction:

x = x₀ + v₀ₓ t + ½ aₓ t²

500 = 0 + (v₀ cos 20°) t + ½ (0) t²

500 = v₀ cos 20° t

Divide the first equation by the second equation.

(5t² − 50) / 500 = tan 20°

5t² − 50 = 500 tan 20°

t² − 10 = 100 tan 20°

t² = 10 + 100 tan 20°

t ≈ 6.81 seconds

Plug into either equation.

v₀ = 500 / (t cos 20°)

v₀ ≈ 78.1 m/s

Rounded to one significant figure, the initial velocity is 80 m/s.