03 · 18 min
Energy, work and power
Kinetic and gravitational stores, work, power, and efficiency.
E = (1/2)mv²
The idea
Energy is transferred or stored. The stores you meet first are kinetic, E = (1/2)mv², and gravitational potential, E = mgh. If friction and air resistance do nothing, mechanical energy is conserved: a loss in one store matches a gain in the other.
Work done is energy transferred by a force moving through a distance: W = Fs, with s along the force. Power is how fast that transfer happens: P = E / t. A watt is one joule per second.
Efficiency compares the useful store or output with the total input. Multiply the fraction by 100 if the answer is wanted as a percentage. Energy is never “used up”; the wasted part is usually heating.
Formulas
Kinetic energy
Ek = (1/2)mv²
The energy of a moving mass. Doubling the speed multiplies the energy by four.
- Ek
- kinetic energy
- J
- m
- mass
- kg
- v
- speed
- m/s
Gravitational potential energy
Ep = mgh
The change when a mass moves vertically through height h. Zero can be anywhere convenient.
- Ep
- gravitational energy
- J
- m
- mass
- kg
- g
- gravitational field strength
- N/kg
- h
- height change
- m
Work done
W = Fs
A force moving through a distance along its own direction. The unit is the joule.
- W
- work done
- J
- F
- force
- N
- s
- distance along the force
- m
Power
P = E / t
The rate of energy transfer. Also P = W / t and, for a steady force, P = Fv.
- P
- power
- W
- E
- energy transferred
- J
- t
- time
- s
Efficiency
efficiency = (useful / total) × 100%
Both energies must be in the same unit. The answer here is a percentage.
- η
- efficiency
- %
- useful
- useful energy output
- J
- total
- total energy input
- J
Worked examples
A falling stone
A 0.40 kg stone is dropped from rest through 5.0 m. Take g = 10 N/kg and ignore air resistance. Find the speed as it reaches the ground.
- Mass
- 0.40 kg
- Height
- 5.0 m
- g
- 10 N/kg
Try it on paper first. Then open the working one step at a time.
Power of a lifting motor
A motor lifts a 50 kg load through 4.0 m at constant speed in 10 s. Take g = 10 N/kg. Find the useful power.
- Mass
- 50 kg
- Height
- 4.0 m
- Time
- 10 s
Try it on paper first. Then open the working one step at a time.
Checks
A 3.0 kg trolley moves at 4.0 m/s. What is its kinetic energy, in joules?
A 60 W lamp is left on for 5.0 minutes. How much energy does it transfer, in joules?
A machine takes in 500 J and transfers 350 J usefully. What is its efficiency, as a percentage?
A motor lifts a load at constant speed. The useful energy stored in the load is mainly
Easy places to lose a mark
- Forgetting to square the speed in kinetic energy.
- Leaving minutes as minutes when power is in watts. Watts already mean joules per second.
- Quoting efficiency as 0.7 when the question asks for a percentage, or the other way round.
Practical
Drop tower
Watch gravitational energy become kinetic energy as a mass falls.