If you lived in a science-fiction movie, one of your biggest worries might be getting crushed by a giant monster. Movie monsters often stand tens or hundreds of meters tall and can crush a building with a single step. They're portrayed as proportionally scaled-up versions of normal creatures. But what would happen in real life if an existing creature was scaled up?

Scaling something up proportionally means that all its lengths are increased by the same factor. For example, if you scaled up a rectangle by \(2\textrm{x},\) then both its width and height would increase by a factor of \(2.\) But scaling something up proportionally doesn't mean all of its geometrical aspects increase identically: for a 3D object, areas will increase at a faster rate than lengths, and volume will increase faster than areas.

Inherent properties like density (the measure of a material's mass per unit of volume) don't change when something is scaled up. Scaling something up so that its length is increased by \(10\textrm{x}\) will increase its volume by \(1000\textrm{x},\) and since its density stays the same, that means its mass is also increased by \(1000\textrm{x}.\)

While mass depends on volume, many characteristics depend on area:

- The amount of force that a muscle can exert depends on its cross-sectional area.
- The amount of heat that skin can transfer to the environment depends on its surface area.
- The amount of weight that a bone can support depends on its cross-sectional area.

Some properties depend on volume, some depend on area, and some depend on length. This mismatch between properties means that any scaled-up creature will run into problems. For example, a scaled-up creature might overheat, or its bones wouldn't support its own weight. Today's challenge looks at how high a scaled-up creature could jump.

One way to estimate how high something can jump is to use the simplified picture of the jumper's muscles pushing up with a force of \(F\) while lifting themselves a distance of \(L\). They achieve their maximum velocity \(v_\textrm{max}\) as they lose contact with the ground and can't apply a force against it anymore.

If the jumper applies a constant force, then their \(v_\textrm{max}\) can be calculated as \[v_\textrm{max} = \sqrt{\frac{2FL}{m}},\] where \(m\) is the jumper's mass. The greater \(v_\textrm{max}\) is, the higher their jump will be. More quantitatively, the height of the jump is proportional to the max velocity squared: \[h \propto v_\textrm{max}^2.\]