Problem — Sci-fi physics, g doubled

Problem

Sci-fi physics: $g$ doubled. What changes for a high jumper, for a free fall from $10\;\text{m}$, and for a ball thrown horizontally, if $g$ suddenly doubled?

Solution

The guiding idea is to write each quantity as a function of $g$ and see how it scales under the substitution $g \to 2g$.

High jumper. With the same push from the legs the body leaves the ground with the same vertical velocity $v_0$; the height reached is $h_\text{max} = v_0^2/(2g)$, so $h_\text{max} \propto 1/g$. Doubling $g$ halves the jump height: someone who jumped $130\;\text{cm}$ would only manage $65$.

Free fall from $10\;\text{m}$. The fall time is $t = \sqrt{2h/g}$, so $t \propto 1/\sqrt{g}$. Doubling $g$ reduces the time by a factor of $\sqrt{2}$: the fall is quicker. The impact velocity $v = \sqrt{2gh}$, instead, grows by a factor of $\sqrt{2}$.

Ball thrown horizontally. With horizontal velocity $v$ and starting height $h$, the flight time is $t = \sqrt{2h/g}$ and the range is $R = v\,t \propto 1/\sqrt{g}$. Doubling $g$ reduces the range by a factor of $\sqrt{2}$ (about $71\%$ of the original): the ball falls sooner and lands closer.

In summary: halved jumps, shorter but more violent falls, shorter trajectories. Everyday life would become decidedly harder.

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Topics: Cinematica Concepts: Caduta libera Skills: Analisi di casi limite e fantafisica