AS Daily A Level Physics question
On level ground with no air resistance, a student launches a foam dart at speed u at 45° to the horizontal. She wants the new launch to have the same total time in the air as before but to land 50% further away horizontally. Which adjustment achieves this most closely?
Answer
The correct answer is C.
Correct: C — Increase the speed by about 28% and lower the launch angle to about 34°. Keeping time the same means the initial vertical component must be unchanged; increasing range by 50% with the same time means the horizontal component must be 1.5 times larger. From u′sinθ′ = u·sin45° and u′cosθ′ = 1.5u·cos45°, tanθ′ = (u′sinθ′)/(u′cosθ′) = 1/1.5 = 2/3, so θ′ ≈ 34°, and u′ ≈ (sin45°/sin34°)u ≈ 1.28u (about a 28% increase). A The speed is unchanged, so lowering the angle reduces the vertical component and shortens the time in the air; the range does not increase by 50% either. B Increasing speed at the same angle scales both components by 1.5, so the time in the air increases by 50% and the range increases by about (1.5)^2 = 2.25 times, not by 50%. C With the vertical component held constant (same time) and the horizontal component 1.5 times larger, the 28% speed increase and ≈34° angle satisfy both conditions. D Raising the angle increases the vertical component, so the time in the air becomes longer (about 50% longer here), violating the "same time" requirement, even though the range change may seem large.