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 some “absolute
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 passes – if 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 position – the potential energy.
So all we need is to find the change in kinetic and potential energies over time, and add
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\h– that is, potential energy is the negative of the force
applied times the change in position.
This is where the “there’s no absolute time” part comes into play – you can’t
have potential energy for a force that changes over time.
And just to be clear, “no changing over time” doesn’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
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.
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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.