Transformation and conservation of energy

A ball that falls

Hold a ball up high and let go. At the top it has potential energy and no kinetic energy. As it falls it loses height and gains speed: its potential energy goes down while its kinetic energy goes up. Just before it lands, nearly all of it is kinetic. The energy has changed form.

The law of conservation of energy

Energy can change from one form to another. It cannot be made or destroyed. The total before and after a change is the same.

For a falling ball, with no air resistance,

At the ground all of $mgh$ has become $\tfrac{1}{2}mv^2$:

The mass cancels: a heavy ball and a light ball land at the same speed.

A swinging pendulum

At the ends of its swing the bob stops for a moment: all potential energy. At the bottom it is fastest: all kinetic energy. In between it has some of each, and the total stays the same.

You should find that the two coloured bars trade places twice in each swing, while the total bar never moves. A longer string swings more slowly, but the bars still add up.

Other changes of energy

• A hydroelectric station: potential energy of the water, then kinetic, then electrical energy.
• A person: chemical energy of food, then kinetic energy of the muscles and heat.
• A cycle dynamo: kinetic energy, then electrical energy, then light.

Where does the energy go in real life?

A dropped ball never bounces back to its starting height, and a pendulum slowly stops. Energy is not disappearing: friction and the air change some of it into heat and sound. Count those, and the total is still the same.

Watch out. $v = \sqrt{2gh}$ uses the height fallen, not the height still left to fall.

A worked example

A mango of $0.25\ \text{kg}$ drops from a branch $4.05\ \text{m}$ high. Find its speed just before it lands, and check the energy.

1. Energy at the top. $PE = 0.25\ \text{kg} \times 9.8\ \text{m/s}^2 \times 4.05\ \text{m} = 9.92\ \text{J}$, and $KE = 0\ \text{J}$.
2. The speed. $v = \sqrt{2 \times 9.8\ \text{m/s}^2 \times 4.05\ \text{m}} = \sqrt{79.4\ \text{m}^2/\text{s}^2} = 8.91\ \text{m/s}$.
3. Check. $KE = \tfrac{1}{2} \times 0.25\ \text{kg} \times (8.91\ \text{m/s})^2 = 9.92\ \text{J}$: all the potential energy.

Practise with new numbers

Try it in Python

The program follows a falling ball and adds up its energy at six heights.

Remember these

• Energy changes form, but it is never made or destroyed.
• With no air or friction, $PE + KE$ stays the same.
• After a fall of $h$, $v = \sqrt{2gh}$, whatever the mass.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A ball falls. Which form of energy grows, and which one shrinks?

2. A stone is dropped from $5\ \text{m}$. How fast is it moving when it lands?

3. Where is a pendulum bob fastest, and where does it stop for a moment?

4. A bouncing ball rises a little less after each bounce. Is energy destroyed?