Two blocks start from the same height, one sliding down a smooth incline and the other in free fall. Which reaches the ground first?

Two blocks start from the same height, one sliding down a smooth incline and the other in free fall. Which reaches the ground first?

• A. Both hit ground simultaneously
• B. Block B reaches ground earlier
• C. Block A reaches ground earlier
• D. Block B never reaches ground

Answer: (B)

The question tests how motion constraints affect the time to descend a height. For the block sliding on a smooth incline, the acceleration along the incline is a = g sin θ, where θ is the incline angle. It starts from rest and must cover a path length L to drop the height h, with L = h / sin θ. The time to slide down is t_slide = sqrt(2L / a) = sqrt(2 (h / sin θ) / (g sin θ)) = sqrt(2h / g) / sin θ. The block in free fall, from the same height, has t_fall = sqrt(2h / g). Since sin θ ≤ 1, t_slide ≥ t_fall, with equality only if θ = 90°, meaning the incline is vertical (which is just free fall). In all typical incline cases, sin θ < 1, so the sliding block takes longer to reach the ground than the freely falling block, even though energy conservation ensures both reach the bottom with the same final speed v = sqrt(2gh) when there’s no friction. Therefore, the block in free fall reaches the ground first.

The question tests how motion constraints affect the time to descend a height. For the block sliding on a smooth incline, the acceleration along the incline is a = g sin θ, where θ is the incline angle. It starts from rest and must cover a path length L to drop the height h, with L = h / sin θ. The time to slide down is t_slide = sqrt(2L / a) = sqrt(2 (h / sin θ) / (g sin θ)) = sqrt(2h / g) / sin θ. The block in free fall, from the same height, has t_fall = sqrt(2h / g).

Since sin θ ≤ 1, t_slide ≥ t_fall, with equality only if θ = 90°, meaning the incline is vertical (which is just free fall). In all typical incline cases, sin θ < 1, so the sliding block takes longer to reach the ground than the freely falling block, even though energy conservation ensures both reach the bottom with the same final speed v = sqrt(2gh) when there’s no friction.

Therefore, the block in free fall reaches the ground first.