Comment by photon_lines

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Comment by photon_lines

So I can show you why it works algebraically -- I'm not sure if it will help you though.

Kinetic energy is the work needed to accelerate something from rest to speed v (velocity). Below we'll use the regular derivative representation to represent an tiny change in a value using notation such as dE (tiny change in energy).

Work can be described as dE (tiny change of energy) = F (force) * dx (tiny change in position). -> dE = F * dx

Force though changes momentum and algebraically can be described as:

F (force) = dp/dt (tiny change in momentum over tiny change in time)

The distance travelled over a infinitesimally small time frame is:

dx (tiny change in position) = v (velocity) * dt (tiny change in time) -> dx = v * dt

So now we have:

dE = F * dx (see our definition above) = dp/dt * v * dt (due to F = dp/dt and x = vt) = v dp (the dt terms cancel each other out and we are left with velocity and rate of change in momentum) -> dE = v * dp

So we are therefore left with:

dE (tiny change in energy) = v (velocity) * dp (tiny change in momentum) -> dE = v * dp

We can translate this as stating: the energy cost of adding a tiny bit of momentum depends on how fast the object is already moving. Now we also know that algebraically:

p (momentum) = m (mass) * v (velocity) -> p = mv

Using derivatives we also have:

dp (tiny change in momentum) = m (mass) * dv (tiny change in velocity) -> dp = m * dv

Going back to our original equation, we now have:

dE (tiny change in energy) = v * dp = v * m * dv (since dp = m * dv)

To get the total energy that it takes to move a mass m from 0 to velocity v, we need to add up all the tiny energy costs from 0 to v, and for this we use the integral to get:

E = ∫(from 0 to v) add up m (mass) * v (velocity) * dv (rate of change in velocity) = 1/2mvv (basic integration) = 1/2m*v^2

So from the above we can see that the kinetic energy rises quadratically with velocty. Now the 'why' this is encoded in the universe: read a bit more about symmetry and Emmy Noether. The laws of physics do not care where you are, when you are, or whether you are moving at a constant speed in a straight line. This is called Galilean symmetry in ordinary classical mechanics. Because the laws of motion have to stay consistent under changes of reference frame, energy cannot just be proportional to velocity. A linear energy law would break the symmetry.

Replies photon_lines · 2026-06-28
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