Calculating changes in energy

P1.2 is the calculation engine of the energy topic. Four equations come up again and again. You must memorise the first three; the specific heat capacity equation is on the AQA equation sheet.

Kinetic energy

$E_k = \tfrac{1}{2} m v^2$

• $E_k$ in joules (J), $m$ in kilograms (kg), $v$ in metres per second (m/s).
• Doubling the speed quadruples the KE — this is why high-speed crashes are so dangerous.

Worked example. A 1500 kg car travels at 20 m/s. Its kinetic store is

$E_k = 0.5 \times 1500 \times 20^2 = 0.5 \times 1500 \times 400 = 300{,}000\text{ J} = 300\text{ kJ}$.

Gravitational potential energy

$E_p = m g h$

• $g = 9.8\text{ N/kg}$ near Earth's surface (use 9.8 or 10 as stated by the question).
• $h$ is the vertical height change, not the distance walked along a slope.

Worked example. A 60 kg climber ascends 12 m. Increase in GPE store:

$E_p = 60 \times 9.8 \times 12 = 7056\text{ J} \approx 7060\text{ J}$.

Elastic potential energy

$E_e = \tfrac{1}{2} k e^2$

• $k$ = spring constant (N/m); $e$ = extension (m).
• The factor of one-half is essential — examiners know students forget it.

Worked example. A spring with $k = 200$ N/m is stretched by 0.05 m.

$E_e = 0.5 \times 200 \times 0.05^2 = 0.5 \times 200 \times 0.0025 = 0.25\text{ J}$.

Specific heat capacity

$\Delta E = m c \Delta\theta$

• $c$ = specific heat capacity (J/kg °C). Water $c \approx 4200$, aluminium $\approx 900$, copper $\approx 385$.
• $\Delta\theta$ = temperature change in °C (the change is the same in K).

Worked example. Heating 0.5 kg of water from 20 °C to 80 °C requires:

$\Delta E = 0.5 \times 4200 \times 60 = 126{,}000\text{ J} = 126\text{ kJ}$.

Linking equations together

Many exam questions chain the equations using conservation of energy:

"A ball of mass 0.2 kg falls from a height of 5 m. Find its speed just before it hits the ground (ignore air resistance)."

GPE lost = KE gained:

$mgh = \tfrac{1}{2}m v^2 \Longrightarrow v = \sqrt{2 g h} = \sqrt{2 \times 9.8 \times 5} \approx 9.9\text{ m/s}.$

Notice the mass cancels — that's why all objects fall at the same speed in vacuum.

Common pitfalls

1. Using cm or grams. SI only — convert before substituting.
2. Forgetting to square v or e. Without the squared term you'll be a factor of 100 out.
3. Vertical vs slope distance — GPE only uses the vertical height change.
4. Dropping the ½ in KE or EPE.
5. Using temperature instead of change: you need $\Delta\theta$, not $\theta$.