Ranking of ground-impact speeds

Problem

Ranking: ground-impact speed in free fall. Same heights as the previous problem: (A) $h = 0{,}5\;\text{m}$; (B) $h = 2\;\text{m}$; (C) $h = 5\;\text{m}$; (D) $h = 20\;\text{m}$. Order the impact speeds at the ground from smallest to largest.

Solution

For free fall from rest, the impact speed is obtained from the relation linking speed and displacement in uniformly accelerated motion ($v^2 = 2gh$):

$v = \sqrt{2\,g\,h}$

The speed also increases with height (monotonically), so the order of the speeds follows that of the heights. Let us calculate with $g \approx 9{,}81\;\mathrm{m/s^2}$:

$v_A = \sqrt{2 \cdot 9{,}81 \cdot 0{,}5} \approx 3{,}13\;\mathrm{m/s}, \quad v_B = \sqrt{2 \cdot 9{,}81 \cdot 2} \approx 6{,}26\;\mathrm{m/s}$

$v_C = \sqrt{2 \cdot 9{,}81 \cdot 5} \approx 9{,}90\;\mathrm{m/s}, \quad v_D = \sqrt{2 \cdot 9{,}81 \cdot 20} \approx 19{,}81\;\mathrm{m/s}$

As with the times, the speed grows only as $\sqrt{h}$: to quadruple the impact speed requires a height $16$ times greater (indeed $h_D = 40\,h_A$ and $v_D \approx 6{,}3\,v_A$).

$\ev{A < B < C < D \quad (3{,}13 < 6{,}26 < 9{,}90 < 19{,}81\;\mathrm{m/s})}$

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Topics: Cinematica Concepts: Caduta libera Skills: Ragionamento di ranking