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• One of the fundamental principles in modern physics is that there's no absolute time.

• And I'm not even talking about relativity making time go at different rates if you're

• going near light speed – I just mean that any time is as good as any other to set your

• clock to zero.

• The predictions of physics work all the same, and it's not like there's someabsolute

• starting time” – I mean, if there were, time zones wouldn't work!

• In addition to allowing time zones, the fact that there's no absolute time also implies

• the law of conservation of energy.

• Here's a simple proof to show that if a force doesn't depend explicitly on time,

• then that force conserves energy!

• First, conserving energy just means that there's a certain number, called the energy, which

• doesn't change as time passesif you compare the energy at two different times

• you'll see no difference.

• And the total energy of a system is the sum of its energy of motion, or kinetic energy,

• and its energy due to positionthe potential energy.

• So all we need is to find the change in kinetic and potential energies over time, and add

• them up!

• Ok, we know the kinetic energy of an object is half of its mass times its velocity squared.

• So the change in kinetic energy over time is just one half m times the difference of

• the squares of the velocities.

• Some clever algebra can rearrange this expression to become the average velocity times m times

• the CHANGE in velocity.

• But a change of velocity over time is just an acceleration, and mass times acceleration

• is equal to the force on the object.

• So the change in kinetic energy of an object over time is just velocity times force.

• On the other hand, the change in the potential energy of an object is negative the amount

• of work it takes to get the object to its current position from its previous position

• independent of the path taken – that is, potential energy is the negative of the force

• applied times the change in position.

• This is where thethere's no absolute timepart comes into playyou can't

• have potential energy for a force that changes over time.

• And just to be clear, “no changing over timedoesn't mean that an object can't

• experience a changing force along the course of its journey: the force might be different

• at different places, but at any particular place, the force must remain the same.

• Anyway, this is all just to say that the change in potential energy is negative F times the

• change in position.

• The negative comes from the fact that if you let the force push you along your potential

• energy decreases, while if you fight in opposition to the force your potential energy increases.

• So the change in potential energy over time is the negative of the force times the change

• in position over time, but change in position over time is velocity!

• Which means change in potential energy of an object over time is negative velocity times

• force.

• And thus the change in the total energy over time, which is the sum of the changes in the

• kinetic and potential energies over time, is v*F plus negative v*F, which equals zero!

• And zero change in energy over time is precisely conservation of energy!

• All for the same reason that time zones work.

• This video is brought to you in part by Audible.com, the leading provider of audiobooks across

• all types of literature, including fiction, non-fiction and periodicals.

• I'd like to recommend the book Unruly Places: Lost Spaces, Secret Cities, and Other Inscrutable

• Geographies by Alastair Bonnett [called Off the Map in the UK] – it's a tour of all

• sorts of weird, abandoned or disregarded places around the world, from the unrecognized self-declared

• island nation of Sealand to traffic islands, and beyond.

• audible for helping me continue to make these videos.

• As a followup, I'll admit that I skimmed over some details in this derivation and it's

• much easier to do with calculus.

• You just have to know that the time derivative of kinetic energy is m times v times a which

• is F*v, and the time derivative of potential energy is -F times dx/dt, or -F*v, and so

• the time derivative of the total energy, dE/dt, is F*v-F*v = 0, which means energy doesn't

• change over time.

• And as a final followup, the most legit and robust version of this proof as it applies

• to ALL OF PHYSICS is called Noether's theorem, discovered by Emmy Noether in 1915.

• It's beautiful physics.

One of the fundamental principles in modern physics is that there's no absolute time.

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B1 energy potential energy kinetic potential velocity kinetic energy

# A Simple Proof of Conservation of Energy

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Summer posted on 2021/10/08
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