AS Daily A Level Physics question
A small trolley starts from rest and rolls a fixed distance L down a smooth ramp. If the ramp angle to the horizontal is doubled from 30° to 60°, which statement must be true about the time to reach the end? Neglect friction and rolling resistance.
Answer
The correct answer is C.
Correct: C — C The time falls to about 0.76 of the original (around 24% less), because time varies with the inverse square root of sinθ. Taking acceleration along the slope as g sinθ and using a fixed distance from rest gives t ∝ 1/√(sinθ), so t60/t30 = √(sin30/sin60) = √(0.50/0.866) ≈ √0.577 ≈ 0.76; direction of acceleration is down the slope and its magnitude increases with angle. A treats time as inversely proportional to acceleration (or sinθ) rather than to its square root; halving the time would require sinθ to increase by a factor of four, which it does not (and sin60 is not double sin30). B uses the correct trend but the wrong dependence, taking t ∝ 1/sinθ to get about 0.58; the correct square-root dependence gives about 0.76 instead. C matches the required proportional reasoning and numeric estimate. D ignores that the component of weight along the slope increases with angle; in the limiting case θ → 0°, sinθ → 0 so t would grow very large, not remain unchanged.