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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.

  • If you go to audible.com/minutephysics, you can try audible out by downloading a free

  • audiobook of your choice.

  • 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.

  • Again, you can download your free audiobook at audible.com/minutephysics and thanks to

  • 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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