Energy: calculations, conservation and efficiency
Kinetic and gravitational potential energy, conservation of energy, efficiency and power, with worked examples. With an experiment to measure the efficiency of a bouncing ball.
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Kinetic energy: Ek = ½·m·v². Gravitational potential energy (near Earth's surface): Ep = m·g·h. Both are measured in joules (J).[3]
Conservation of energy: in a closed system the total energy is constant. Energy can change form or be transferred, but it is never created or destroyed.[3][4]
Example: a 10 kg rock falls from a 20 m cliff. After falling 10 m it has lost m·g·Δh = 10 × 9.8 × 10 = 980 J of potential energy, so (ignoring air resistance) it has gained 980 J of kinetic energy.[3]
Efficiency = useful energy (or work) output ÷ total energy input. Representative values: electric motor 98%, gas heater 90%, coal power plant 42%, petrol engine 30%, human body cycling 20%, solar cell 10%. The rest is mostly dissipated as thermal energy, for example through friction.[4][3]
Power is the rate of energy transfer: P = E/t, in watts (1 W = 1 J/s). A 60 W incandescent bulb turns only about 5 W into light: an efficiency of about 8%. Electricity bills count energy in kilowatt-hours: power in kW × time in h.[5]
Choosing efficient devices saves energy: LED bulbs use at least 75% less energy than incandescent bulbs, which release about 90% of their energy as heat.[6]
Try it: Efficiency of a bouncing ball
You need
- Two or three different balls (tennis, bouncy, ping-pong)
- Balance
- Metre rule or tape fixed vertically to a wall
- Phone camera (slow motion if possible)
- A hard floor
Safety
Drop the balls, don't throw them. Keep the area clear and don't stand on tables or chairs to drop from higher than you can reach safely.
What to do
- Measure the mass of each ball.
- Drop a ball from h₁ = 1.00 m (bottom of the ball) and film it next to the rule. Read the height h₂ the bottom of the ball reaches after the first bounce.
- Repeat five times and calculate the mean h₂ and its spread (half the range).
- Calculate E_p before the drop (m·g·h₁) and at the top of the bounce (m·g·h₂).
- Efficiency of the bounce = useful energy out ÷ energy in = m·g·h₂ ÷ m·g·h₁ = h₂ ÷ h₁.
- Repeat for the other balls and for two more drop heights. Does the efficiency depend on the drop height?
- Optional: from the video, find the speed just before impact and compare ½·m·v² with m·g·h₁.
What happens, and why?
At the top the ball has gravitational potential energy; just before impact this has become kinetic energy (minus a little lost to air resistance). During the bounce part of the energy is dissipated as thermal energy (and sound), so the ball rises less high. The total energy is conserved, but less of it is useful, so the efficiency h₂/h₁ is below 1. Reading heights from a video limits the precision, which is why repeats and the spread matter.[3][4]
Play: How many joules?
Calculate the energy (g ≈ 9.8 m/s²; E_p = m·g·h, E_k = ½·m·v²) and pick the closest value.
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Quiz
Other ages: Ages 6–7 · Ages 10–11