Question

The severity of a fall depends on your speed when you strike the ground. All factors but the acceleration due to gravity being the same, how many times higher could a safe fall on the Moon be than on Earth (gravitational acceleration on the Moon is about 1/6 that of the Earth)?

Short answer

The item might fall from a height of six times its initial height while remaining safe.

Step-by-step solution

Kinetic energy and potential energy

When an item is released from rest in the open air (with no friction), its speed is entirely determined by gravity's acceleration, g.

If we drop an item of mass m from a height h near the Earth's surface, it has initial mechanical energy as:

$\mathbf{U}\mathbf{=}\mathbf{mgh}$

When an item collides with the ground, all mechanical energy (potential energy) is converted to kinetic energy as:

$\mathbf{K}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{mv}}^{\mathbf{2}}$

Where v denotes the speed of the object shortly before it collides with the earth.

If we know what speed v is safe for the object's integrity, we can determine how high it was dropped.

If the object is on the moon

As described, kinetic energy will be equal to potential energy for the case.

After equating the equation, we can find out the value of h.

This value of height is the safe height.

$\begin{array}{c}\mathrm{mgh}=\frac{1}{2}{\mathrm{mv}}^2\\ \mathrm{h}=\frac{{\mathrm{mv}}^2}{2\mathrm{mg}}\\ \mathrm{h}=\frac{{\mathrm{v}}^2}{2\mathrm{g}}\end{array}$

If the case happened on the moon, then the value would have become,

${\mathrm{h}}_{\mathrm{M}}=\frac{{{\mathrm{v}}_{\mathrm{M}}}^2}{2{\mathrm{g}}_{\mathrm{M}}}$

Now it is very clear that the gravitational acceleration in the moon is 1/6^th times that of the gravitational pull of the earth.

Hence if we put that value of g_m in the equation, we get,

$\begin{array}{c}{\mathrm{h}}_{\mathrm{M}}=\frac{{\mathrm{v}}^2}{2\times \left(\frac{1}{6}\mathrm{g}\right)}\\ {\mathrm{h}}_{\mathrm{M}}=6\left(\frac{{\mathrm{v}}^2}{2\times \left(\mathrm{g}\right)}\right)\\ {\mathrm{h}}_{\mathrm{M}}=6\mathrm{h}\end{array}$

Hence it is clear that an object could fall from six times the original height and still be safe.