Range Safety and Projectile Motion at the Academy Field
A cadet candidate is evaluated on a physics problem during the entrance exam. A small training projectile is launched from ground level at 20 m/s at an angle of 30° above the horizontal over flat terrain. The launch point is 15 m from a safety fence that is 3.0 m high. The candidate is told to ignore air resistance and to take g = 9.8 m/s². The examiner asks the candidate to decide whether the projectile clears the fence on its way up, and to explain what would change if the launch angle were raised to 45° while keeping the same speed. The candidate must justify every numerical claim with a calculation, not with a remembered formula alone.
1. Determine whether the projectile clears the 3.0 m fence located 15 m downrange, and show the reasoning.
Read the worked answer
Step 1 - Components. v₀ₓ = 20·cos30° ≈ 17.32 m/s; v₀ᵧ = 20·sin30° = 10.0 m/s.
Step 2 - Time to reach x = 15 m. t = x / v₀ₓ = 15 / 17.32 ≈ 0.866 s.
Step 3 - Height at that time. y = v₀ᵧ·t − ½g·t² = 10.0·0.866 − 4.9·(0.866)² ≈ 8.66 − 3.67 ≈ 4.99 m.
Conclusion. Since 4.99 m > 3.0 m, the projectile clears the fence by about 2.0 m. The judgement must be based on the height at the fence's horizontal position, not on maximum height, because the fence is a vertical obstacle at a specific distance.
2. If the launch angle is increased to 45° at the same 20 m/s, does the projectile still clear the fence? Explain the trade-off.
Read the worked answer
Step 1 - New components. v₀ₓ = v₀ᵧ = 20·sin45° ≈ 14.14 m/s.
Step 2 - Time to x = 15 m. t = 15 / 14.14 ≈ 1.061 s.
Step 3 - Height at fence. y = 14.14·1.061 − 4.9·(1.061)² ≈ 15.00 − 5.52 ≈ 9.48 m.
Conclusion. It clears the fence by a much larger margin (~9.5 m). The trade-off: a steeper angle gives a higher trajectory but a lower horizontal speed, so the projectile takes longer to reach the fence and reaches a greater height at that distance. The candidate should note that maximum range occurs near 45° only on flat ground with no obstacle; here the obstacle is what matters.