IMAT Physics · Dynamics, Work and Energy

Conservation of Mechanical Energy, Power and Efficiency

Without friction, kinetic plus potential energy stays constant; with friction some becomes heat; power is how fast energy is transferred.

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Why this matters

Two ministry papers asked here: what friction does to the mechanical energy of a body sliding down a slope (2026), and the power of a braking force (2024). The 2016 paper asked which statements about the energy and forces of a ball thrown upwards are true.

Concept 1 of 2: Conservation of mechanical energy, and what friction does

Energy is never made or destroyed, only moved from one store to another. A falling ball swaps potential energy for kinetic energy, joule for joule. When friction or air resistance acts, some of that energy leaves the mechanical stores as heat, so the mechanical energy falls even though the total energy is still conserved.

Definition

Mechanical energy = kinetic + potential energy (gravitational and elastic).
If only gravity (or a spring) does work, mechanical energy is conserved: $\tfrac{1}{2}mu^2 + mgh_1 = \tfrac{1}{2}mv^2 + mgh_2$ .
Falling from rest through a height $h$ : $v = \sqrt{2gh}$ , the same for every mass.
With friction or air resistance, mechanical energy decreases; the loss equals the work done against friction and becomes internal energy (heat).
The total energy, including heat, is always conserved.

Mechanical energy without friction

\tfrac{1}{2}mv^2 + mgh = \text{constant} \qquad v = \sqrt{2gh}\ \text{(from rest)}

$h$ height, in m
$v$ speed, in m/s

Worked example

A 2.0 kg ball is dropped from rest from a height of 5.0 m. Ignoring air resistance and taking

g = 10\ \text{m/s}^2

, how fast is it moving just before it lands? Would a 4.0 kg ball land faster?

Practice this conceptself-check · 4 quick reps

Friction lowers mechanical energy but not total energy

On a rough slope the mechanical energy of a sliding body decreases; it does not stay constant, double or vanish. The missing amount has become heat, so the total energy is still conserved.

Concept 2 of 2: Power and efficiency

Two people can climb the same stairs and do the same work, but the one who runs up has the higher power: power is how quickly work is done. No real machine hands on all the energy it takes in; efficiency is the fraction that comes out as the useful kind.

Definition

Power is work done (or energy transferred) per unit time: $P = W/t$ , in watts, $1\ \text{W} = 1\ \text{J/s}$ .
A force moving its point of action at speed $v$ along the force has power $P = Fv$ .
Efficiency = useful energy (or power) out ÷ total energy (or power) in, often as a percentage. It is always less than 100% in real machines.
The kilowatt-hour is an energy unit: $1\ \text{kWh} = 1000\ \text{W} \times 3600\ \text{s} = 3.6 \times 10^6\ \text{J}$ .

Power and efficiency

P = \frac{W}{t} = Fv \qquad \eta = \frac{\text{useful output}}{\text{total input}} \times 100\%

$P$ power, in W
$W$ work done or energy transferred, in J
$\eta$ efficiency

Worked example

An electric motor lifts a 50 kg load through 12 m in 20 s, drawing 400 W from the mains. Taking

g = 10\ \text{N/kg}

, find the useful power output and the efficiency.

Practice this conceptself-check · 3 quick reps

The kilowatt-hour is energy, not power

A kilowatt is a rate; a kilowatt-hour is an amount of energy,

3.6 \times 10^6\ \text{J}

. Likewise a power needs a time or a speed: a force and a distance alone give work, not power.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Conservation of mechanical energy, and what friction does
Mechanical energy without friction
$\tfrac{1}{2}mv^2 + mgh = \text{constant} \qquad v = \sqrt{2gh}\ \text{(from rest)}$
Power and efficiency
Power and efficiency
$P = \frac{W}{t} = Fv \qquad \eta = \frac{\text{useful output}}{\text{total input}} \times 100\%$

Watch out for (2)

Friction lowers mechanical energy but not total energy
→ Conservation of mechanical energy, and what friction does
The kilowatt-hour is energy, not power
→ Power and efficiency

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