Gravitational potential energy

Lift something up and you fill its gravitational potential store.

gravitational potential energy = mass × gravitational field strength × height
Ep = m g h

Energy in J, mass in kg, g in N/kg, height in m. On Earth, g = 9.8 N/kg: the question will tell you which value to use. Recall it: none of the three boards puts it on the sheet.

  1. A 50 kg climber climbs 12 m. Take g = 9.8 N/kg.
  2. E = 50 × 9.8 × 12
  3. E = 5880 J

To find a height: h = E ÷ (m g). If the energy is in kJ, multiply by 1000 first.

Elastic potential energy

A stretched or squashed spring stores energy, as long as it is not stretched past its limit of proportionality.

elastic potential energy = 0.5 × spring constant × (extension)²
Ee = ½ k e²

k in newtons per metre (N/m), extension e in metres. This one is given on the equation sheet for AQA, Edexcel and OCR.

  1. A spring with k = 400 N/m is stretched 5 cm.
  2. e = 5 cm = 0.05 m
  3. E = ½ × 400 × 0.05² = 0.5 × 400 × 0.0025
  4. E = 0.5 J

Leave e in cm and you get 5000 J instead of 0.5 J. Convert before you square.

Energy is conserved

When nothing is wasted (no air resistance, no friction), energy lost from one store all turns up in another.

  1. A ball is dropped from 5 m. How fast is it going when it lands?
  2. Ep lost = Ek gained: m g h = ½ m v²
  3. The mass cancels: v² = 2 g h = 2 × 9.8 × 5 = 98
  4. v = √98 = 9.9 m/s (2 s.f.)

The same idea works for a swing (kinetic at the bottom becomes gravitational at the top: h = v² ÷ 2g) and for a spring launcher (½ k e² becomes ½ m v²). These two-step questions are usually on Higher papers.

most EpEk = 0 most EpEk = 0 most Ek, least Ep

Watch the stores swap in a pendulum

Set the length to 2 m and the start angle to 40°. Open Advanced, set gravity to 9.8 and turn on Energy bars, then start it. At each end the potential bar is full and the kinetic bar is empty; at the bottom they swap, and the total never changes. The bob drops 2 × (1 − cos 40°) = 0.47 m, so it should reach √(2 × 9.8 × 0.47) = 3.0 m/s at the bottom. Then add some damping: what happens to the total, and where has the energy gone?

Open the Pendulum simulation in a new tab

Energy in a bouncing spring

Set the spring constant to 50 N/m, the mass to 2 kg and the amplitude to 0.4 m, and turn on Energy bars under Advanced. The energy readout is ½ × 50 × 0.4² = 4 J. Check the maximum speed readout (2 m/s): ½ × 2 × 2² is also 4 J. Kinetic at the middle, elastic at the ends, same total. Double the amplitude: by how much does the energy go up?

Open the Mass on a spring simulation in a new tab

Your first potential energy calculation

Write E = m g h, use g = 9.8 N/kg as the question says, then the unit.