It’s comparatively widespread on Physics Boards to see arguments which are successfully just like the next:

After we bounce off the bottom, the bottom doesn’t transfer. Due to this, the pressure from the bottom on us does zero whole work. Because the pressure does no work, we can not achieve any kinetic power. We subsequently can not bounce off the bottom.

Now, the conclusion right here is clearly false. The world excessive bounce document is 2.45 meters, positively bigger than zero. So the place did the power come from? This Perception seeks to make clear this in a reasonably accessible manner.

## An idealized instance

Earlier than leaping off into bipedal mammals competing within the excessive bounce, allow us to take a look at an idealized instance. This instance will assist us perceive what’s going on a bit of higher.

A mass ##m## has a great spring of size ##ell## and spring fixed ##okay## hooked up to it. The mass and spring are pressed towards a set wall such that the spring is compressed by a distance ##D##, see the determine beneath.

In different phrases, the mass of the spring is zero, and the pressure at its ends is given by Hooke’s legislation. All of this happens in a horizontal airplane, which means that we don’t have to cope with gravity.

As soon as launched, the spring pushes the mass away from the wall. Much like the bounce off the bottom, the wall supplies no work. By the identical reasoning as in our instance argument, the mass can not transfer away from the wall.

## The place is the power?

So the place does the power come from? As a result of the spring will definitely launch the mass away from the wall. As a way to reply this, allow us to first take a look at the method of compressing the spring. Particularly, we contemplate a small section of the spring between the coordinates ##x_0## and ##x_0 + Delta x## when the spring is relaxed. The compressing pressure pushing on its ends is ##F = -kappa epsilon## in accordance with Hooke’s legislation (see the determine beneath). Right here ##epsilon## is the pressure and ##kappa = okay ell##.

Altering the pressure by ##depsilon##, the decrease finish of the string section strikes by ##x_0 depsilon## and the higher by ##(x_0+Delta x)depsilon##. The whole work performed on the section turns into $$dW = F x_0 depsilon – F (x_0+Delta x) depsilon = kappa epsilon Delta x , depsilon.$$ Integrating this from no pressure to a pressure ##epsilon_0## results in $$W = kappa Delta xint_0^{epsilon_0} epsilon,depsilon = frac{kappaepsilon_0^2}{2} Delta x.$$ That is the overall power saved within the spring section at pressure ##epsilon_0##.

That the overall power saved within the spring is $$W = frac{kd^2}{2}$$, the place ##d## is the compression of the spring and ##okay = kappa/ell## is the spring fixed, follows instantly from the above.

## Vitality flux

The dialogue above suggests the concept power can enter or exit an object and stay as inside power. This happens by means of forces performing on the thing performing work. A pressure ##vec F## performing on an object over a displacement ##dvec r## will do a complete work of ##vec F cdot dvec r##. Within the instance above, ##x_0 depsilon## replaces ##vec dr## for the decrease finish as that is the decrease finish’s displacement and we work in a single dimension. Equally, we’ve got a directed one-dimensional pressure ##F## as a substitute of the three-dimensional vector ##vec F##.

The amount ##F x_0 depsilon = – kappa epsilon x_0 depsilon## is, subsequently, a measure of the quantity of power flowing upward by means of the spring at place ##x_0## when the pressure modifications by ##depsilon##. When ##epsilon## is unfavourable, i.e., when the spring is compressed, power will movement upward if ##depsilon## is constructive. In different phrases, when the spring is compressing power flows left within the spring and proper when it’s decompressing.

## Launching the mass

The spring will decompress through the launch of the mass. The interior power saved within the spring then flows from the spring into the mass. Denoting the compression of the spring ##D(t)##, we discover that $$D(t) = D_0 cos(omega t)$$ with ##omega^2 = kappa/mell## through the launch, the place the preliminary compression is ##D_0##.

The launch time interval is ##0 leq t leq pi/2omega##. The pressure ##epsilon## is said to ##D## as ##epsilon = -D/ell##. We subsequently get hold of $$frac{depsilon}{dt} = -frac{D'(t)}{ell} = frac{D_0omega sin(omega t)}{ell}.$$

Consequently, the power flowing up by means of the spring at place ##x## is $$frac{dW}{dt} = -kappa epsilon x, depsilon = kappa frac{D_0cos(omega t)}{ell} x frac{D_0omega sin(omega t)}{ell} = frac{kappa D_0^2}{2ell^2} omega xsin(2omega t).$$

It’s pure that this grows linearly with ##x##. As all power launched from the spring flows into the mass, the power movement will get bigger the nearer to the mass we get.

## Relation to the jumper

A jumper’s legs are not at all a great spring. Nonetheless, the dialogue above does give some perception into the difficulty introduced to start with:

- The higher physique will obtain web work from the legs very similar to the mass obtained web work from the spring throughout launch.
- The online work from the bottom is zero.
- The power is supplied from inside power saved within the jumper’s muscle tissues. Simply because the power right here was supplied from inside power saved within the spring.

Some variations are additionally notable:

- Not like the spring, the jumper’s decrease physique may have non-zero kinetic power on the finish.
- Vitality will even be misplaced within the type of warmth because the effectivity of conversion of inside power to macroscopic kinetic power isn’t 100%.

Whereas the bottom doesn’t do work on the jumper, the jumper’s momentum *is* supplied by the pressure from the bottom. This momentum is distributed all through the jumper’s physique by inside forces.

Professor in theoretical astroparticle physics. He did his thesis on phenomenological neutrino physics and is presently additionally working with completely different elements of darkish matter in addition to physics past the Customary Mannequin. Creator of “Mathematical Strategies for Physics and Engineering” (see Perception “The Start of a Textbook”). A member at Physics Boards since 2014.