Power, Efficiency, and Where Energy Really Goes

Power: the rate of energy transfer

Doing a job is one thing; doing it quickly is another. Power measures how fast work is done or energy is transferred. Two cranes can both lift a load to the same height (same work), but the more powerful one does it in less time.

P = \frac{W}{t}, \qquad P = \frac{dW}{dt}

Power is work per unit time — the average form and the instantaneous form.

1\ \text{W} = 1\ \text{J/s}, \qquad 1\ \text{hp} \approx 746\ \text{W}

The unit of power is the watt; one horsepower is about 746 watts.

P = F v

For a force pushing an object moving at speed v, power equals force times velocity.

Worked example: a 50 kg student runs up a 3.0 m flight of stairs in 4.0 s. Their power output against gravity is P = mgh/t = (50)(9.8)(3.0)/4.0 \approx 368 W — about half a horsepower, sustained only briefly.

When friction acts: energy dissipation

Once a nonconservative force like kinetic friction enters, mechanical energy is no longer conserved. It does not vanish — it converts into thermal energy, warming the surfaces in contact. This is energy dissipation.

W_\text{friction} = -f_k\, d, \qquad E_\text{thermal} = f_k\, d

Friction does negative work on the object; the same amount of energy appears as heat.

\tfrac{1}{2} m v_i^2 + m g h_i = \tfrac{1}{2} m v_f^2 + m g h_f + f_k d

Full energy accounting: initial mechanical energy equals final mechanical energy plus the heat generated by friction.

The grand law and efficiency

Widen the ledger to include heat, sound, light, and every other form, and you recover the unbreakable law of conservation of energy: total energy is never lost, only converted. Energy that seems to 'disappear' has simply become a form we were not tracking. Real machines convert only part of their input into useful output; the rest leaks away, mostly as heat. We quantify this with efficiency.

\eta = \frac{\text{useful energy out}}{\text{total energy in}} \le 1

Efficiency is the fraction of input energy that comes out as useful work — always less than 100% for real machines.

Worked example: putting it all together

A 0.50 kg block starts from rest and slides down a frictionless ramp from a height of 2.0 m. At the bottom it slides onto a rough horizontal floor with coefficient of kinetic friction \mu_k = 0.30. How far does it travel before stopping? We chain two ideas: conservation of mechanical energy on the ramp, then the work-energy theorem on the floor.

KE_\text{bottom} = m g h = (0.50)(9.8)(2.0) = 9.8\ \text{J}

Step 1 — all the potential energy from the drop becomes kinetic energy at the bottom.

f_k = \mu_k m g = (0.30)(0.50)(9.8) = 1.47\ \text{N}

Step 2 — the kinetic friction force on the flat floor.

d = \frac{KE_\text{bottom}}{f_k} = \frac{9.8}{1.47} \approx 6.7\ \text{m}

Step 3 — friction must dissipate all 9.8 J, so it acts over 6.7 m before the block halts.

Where energy leads next

The energy ledger keeps growing as your physics does. In collisions, kinetic energy may or may not be conserved (elastic versus inelastic), while momentum always is. In thermal physics, the heat that friction produced becomes a subject in its own right, governed by the laws of thermodynamics. And at the deepest level, Einstein showed that mass itself is a stored form of energy — mass-energy equivalence.

E = m c^2

The ultimate entry in the energy ledger: mass is energy, with c² as the (enormous) exchange rate.

You now hold one of the most powerful tools in all of physics. Many problems that would be a nightmare with forces and calculus fall in a line or two once you ask the simple question: where did the energy start, and where did it go?