Conservation of energy

The principle, and the two big stores

The principle of conservation of energy says energy cannot be created or destroyed, only transferred from one store to another. “Using” energy means moving it somewhere less useful. The total never drops. Mechanics runs almost entirely on two stores, the first being kinetic energy, the energy of motion.

The second is the change in gravitational potential energy when a mass moves through a height.

Respect the square in E_k. Double the speed and the kinetic energy goes up four times, which is the physics sitting underneath every stopping-distance table you were shown at GCSE.

Where the two formulas come from

Neither equation is handed down from nowhere. Both are the work done by a force, W = Fs$W = Fs$W = Fs, worked through to its destination, and both derivations are short enough to be worth carrying rather than trusting.

Take the potential energy first. To raise a mass m through a height Δh without giving it any speed, you pull upward with a force equal to its weight, mg, and that force acts along the displacement, so W = Fs becomes W = mg × Δh with no angle to worry about. The work you put in is held in the gravitational field and returned in full if the mass is released, so the store gains exactly the work done, ΔE_p = mgΔh$\Delta E_{p} = mg\Delta h$ΔE_p = mgΔh. One condition hides inside treating g as a single number, which is that the field must be uniform over the height climbed. Near the Earth's surface it is, and over the height of a mountain it is not quite.

Kinetic energy needs one extra ingredient, the equations of motion. Push a mass m from rest with a constant resultant force F through a distance s. The work done on it is W = Fs$W = Fs$W = Fs, and Newton's second law turns the F into ma, so W = mas. Now bring in v² = u² + 2as$v^{2} = u^{2} + 2as$v^2 = u^2 + 2as with u = 0, which leaves 2as = v²$2as = v^{2}$2as = v^2, so the product as on its own is 12v²$\tfrac{1}{2}v^{2}$(1/2)v^2. Put that back in place of as, and the work done comes out as W = m × ½v² = ½mv².

Nothing was lifted and nothing was rubbed, so every joule of that work went into the motion. The energy a mass m carries by virtue of moving at speed v is therefore E_k = 12mv²$E_{k} = \tfrac{1}{2}mv^{2}$E_k = (1/2)mv^2. The ½ has a definite source, and it is the 2 in 2as. It is not a fudge factor, and a student who has watched it arrive can rebuild the formula when memory fails.

Frictionless exchanges

With no resistive forces, the two stores simply trade. A pendulum converts potential to kinetic on the way down and back again on the way up; a thrown ball does the same in one exchange.

That exchange opens a fast route through problems which would be tedious with forces. Equate mgΔh to ½mv² and the speeds and heights follow with no force diagram at all. The mass usually cancels, so a heavy pendulum and a light one released from the same height reach the bottom at identical speed.

When friction dissipates energy

Real systems leak, and the leak has a name, work done against resistive forces. Nothing is destroyed. The energy goes to internal energy of the surfaces and the air, warming both. Conservation then reads as a budget with three lines in it.

Energy in equals useful energy out plus energy to the resistive forces, every joule accounted for. Questions run both ways. Give you two lines and ask for the third, or give you all three and ask whether the energy balances. One sentence handles either. Write the stores at the start, write the stores at the end, and let the difference be the work done against resistance.