How Can We Jump When the Ground Does No Work?

[Orodruin]

It is relatively common on Physics Forums to see arguments that are effectively similar to the following:
When we jump off the ground, the ground does not move. Because of this, the force from the ground on us does zero total work. Since the force does no work, we cannot gain any kinetic energy. We therefore cannot jump off the ground.
Now, the conclusion here is false. The world high jump record is 2.45 meters, definitely larger than zero. So where did the energy come from? This Insight seeks to clarify this in a fairly accessible way.

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ToggleAn idealized exampleWhere is the...

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[nasu]

You start with a horizontal spring but in the analysis you refer to "lower end" and energy flowing upward. It may be confusing.

 

[Orodruin]

You start with a horizontal spring but in the analysis you refer to "lower end" and energy flowing upward. It may be confusing.

I’ll add ”in the figure”

 

[Delta2]

This is one of the "rare" (which is not so rare after all but it is rare regarding to what is being taught and problems solved in introductory physics) cases in physics where the force that changes the momentum is not the same with the force that changes the kinetic energy. In this example the force from the ground changes the momentum , but internal forces in the knee and in the torso change the kinetic energy.

 

[DrClaude]

Wouldn't the explanation be simpler with a spring that is not attached to the wall (so no oscillation)?

 

[Delta2]

Wouldn't the explanation be simpler with a spring that is not attached to the wall (so no oscillation)?

Nope. From conservation of momentum, the body can't attain substantial momentum because the spring has negligible mass hence it has negligible momentum after the decompression happens.

 

[DrClaude]

Nope. From conservation of momentum, the body can't attain substantial momentum because the spring has negligible mass hence it has negligible momentum after the decompression happens.

Sorry, but I don't see how the spring being attached or not to the wall changes anything in this respect.

 

[Delta2]

Sorry, but I don't see how the spring being attached or not to the wall changes anything in this respect.

If the spring is not attached to the wall, there is no external force to the spring+mass system so its momentum is conserved.

 

[jbriggs444]

If the spring is not attached to the wall, there is no external force to the spring+mass system so its momentum is conserved.

The spring is not being de-coupled from the wall entirely. It is still prevented from interpenetrating. It can push. But it is detached so that it cannot pull. It is not glued, nailed, screwed or otherwise fastened down.

 

[Delta2]

The spring is not being de-coupled from the wall entirely. It is still prevented from interpenetrating. It can push. But it is detached so that it cannot pull. It is not glued, nailed, screwed or otherwise fastened down.

Ok I see it can push but not pull, in that case momentum is not conserved and yes we will not have oscillation.

 

[Orodruin]

Wouldn't the explanation be simpler with a spring that is not attached to the wall (so no oscillation)?

The spring is not attached to the wall for this very reason. I might have made it clearer in the first image though. Also note that I have been careful to only discuss the launch up to the time when the spring is back to its unloaded length.

 

[Orodruin]

If the spring is not attached to the wall, there is no external force to the spring+mass system so its momentum is conserved.

It does not need to be physically fixed to the wall to compress the spring.

 

[MevsEinstein]

In the article, it says that the spring doesn't have mass. So what would the spring be in real life? I'm confused since I am still a middle schooler.

 

[Orodruin]

It is what we call an idealised spring. It does not exist in reality. It is a way to be able to neglect certain aspects in order to focus on the relevant issues in a physics problem while still getting a reasonably accurate result for many situations where you have a spring that is close to being ideal. An ideal spring has no mass and responds to compression and elongation with a linear force. A real life spring does neither of those except approximately.

 

[Delta2]

Btw @Orodruin implies that our legs play the role of the spring as we first bend them and lower our self down (compression of spring) and then we de -bend them (sorry my english fail here can't find the proper word) in order to jump high.

 

[Orodruin]

Effectively it is a description of how stored energy can be released without the ground doing any work. To use an idealised spring is a way to be able to focus on how the energy flows in the system without the complications of needing to also discuss the kinetic energy in the launch mechanism. Similarities and differences in the case of a jump are discussed at the end.

 

[MevsEinstein]

It is what we call an idealised spring. It does not exist in reality. It is a way to be able to neglect certain aspects in order to focus on the relevant issues in a physics problem while still getting a reasonably accurate result for many situations where you have a spring that is close to being ideal. An ideal spring has no mass and responds to compression and elongation with a linear force. A real life spring does neither of those except approximately.

So is it like those physics problems that say "calculate acceleration neglecting air resistance"?

 

[Orodruin]

Yes. Unlike mathematics, a big part of physics is about making descriptions of real situations that may not be completely accurate but are accurate enough to describe what you want at a sufficiently precise level.

 

[Orodruin]

Another reason to deal with idealisation is to be able to highlight certain aspects of a problem without running into unnecessary complications. This is the case here. What is being discussed is how a force from the ground can lead to an object moving without providing any work within the context of classical mechanics. The example highlights this.

 

[DoctorHoon]

So I'm digging WAY back here to high school and college physics, but in classical mechanics, doesn't the Earth technically "move"? Equal and opposite sort of thing? It's just that the mass of Earth is so huge in relation to the human that it's negligible? Sorry if I'm missing something...

Terrific forum and post, BTW!

 

[jbriggs444]

in classical mechanics, doesn't the Earth technically "move"? Equal and opposite sort of thing? It's just that the mass of Earth is so huge in relation to the human that it's negligible?

Yes, you are right. We are agreeing to neglect that.

 

[Orodruin]

So yes, as @jbriggs444 said we are making the approximation that the Earth is stationary. However, since the Earth is pushed down, the miniscule energy that is transferred across the jumper-Earth contact area would actually flow from the jumper and into the Earth so this would make the misconception even more puzzling. The same analysis applies however and the energy released in the jump starts as internal energy in the jumper’s muscles.

Edit: We are also completely sweeping the entire issue of using other inertial frames under the carpet. Opening that is an entirely different can of worms…

 

[Nik_2213]

IIRC, enthusiastic fans jumping up and down in unison at stadia routinely register on nearby seismo' sensors...

Yup, there's definitely energy transfer...

( Don't forget the thunderous 'Thumper' trucks used for geo-surveys... )

 

[Orodruin]

IIRC, enthusiastic fans jumping up and down in unison at stadia routinely register on nearby seismo' sensors...

Sure, but as mentioned above that transfer is generally into the ground because the floor is not entirely rigid. I also strongly suspect that most of the energy transfer occurs on the landing *thump* rather than at the jumpoff.

( Don't forget the thunderous 'Thumper' trucks used for geo-surveys... )

We have also all seen Dune … Thumpers can be very useful when dealing with enormous sandworms.

 

[OPhy]

Think about a spring in outer space that had a device attached to it that used electricity to contract it together. When the spring is contracted, the electrical energy is converted to potential energy, then that's converted to kinetic energy when the spring is released, which leads the spring to be too stretched out, which leads it to contract, etc. So in outer space, a self-contracted spring will oscillate forever (assuming no losses).

Now put this spring against a wall of very large mass. The electrical energy contracts the spring, but now assume that the spring is contracted in a way that keeps it touching the wall with the help of gravity. So when the spring is fully contracted with it's edge on the wall, it begins to stretch out. However, it can't stretch "through" the wall, so it ends up exerting a force on the wall you can work out using Hooke's law (assuming ideal spring). By Newton's Third Law, that force ends up pushing the spring upwards, and if it's large enough, it will overcome the gravity of the wall and launch away.

Now what about conservation of momentum? There is no contradiction here. Throw a ball at a heavy wall, ball goes in one direction, hits wall, changes direction. How does that preserve conservation of momentum? Well the wall is heavy, so the infinitesimal velocity it moves at away from the wall gets multiplied by it's humongous mass and you get a finite quantity that compensates the ball changing direction. Same thing here.

 

[rcgldr]

The force results in an equal in magnitude but opposite in direction change in momentum of jumper and Earth away from the center mass of Earth and jumper. Due to the Earth being massive, the change in speed of the Earth is extremely tiny, so almost all of energy goes into the jumper.

 

[Orodruin]

The insight mentions the jumper's lower body doesn't move,

No it doesn’t. In fact, the point being made is that the lower body extending (although in the example idealised by a spring) is what gives an energy flow into the body from the energy release in the lower body.

Edit: In particular, the Insight explicity states
“Unlike the spring, the jumper’s lower body will have non-zero kinetic energy at the end.”
The zero energy of the spring results from the spring in the example being an idealised massless spring. This is in order to simplify the considerations but does not affect the main point.

 

[rcgldr]

“Unlike the spring, the jumper’s lower body will have non-zero kinetic energy at the end.”

I thought I removed that paragraph, but lost an edit. It's fixed now. I'll delete this post later, as it was just to acknowledge your post and that my prior post is now corrected.

 

[Orodruin]

The force results in an equal in magnitude but opposite in direction change in momentum of jumper and Earth away from the center mass of Earth and jumper. Due to the Earth being massive, the change in speed of the Earth is extremely tiny, so almost all of energy goes into the jumper.

See #20-#22. The misconception that is being addressed is that the ground not doing work on the jumper would prevent kinetic energy buildup. This misconception is solved by showing how internal energy can be released by parts of the body moving at different velocities. Due to the small motion of the Earth (neglected in the Insight), the energy flow into the Earth is small (and amounts to the ground doing negative work on the jumper).

 

[Swamp Thing]

Interesting article, very nicely written.

But here's a thought: The main target audience is people who actually ask the question posed in the title. I know that when I was at the stage of asking such questions, I would have been scared off if I saw any calculus in the article. I would have engaged more readily with pictures (maybe animated gifs?) and simple pre-calculus explanations.