Exercises: Energy transformations and conservation of energy

An apple detaches from a tree and falls to the ground, as shown in the diagram.

1. a) What happens to its potential energy during the fall?
2. b) And to its kinetic energy?
3. c) Explain the energy transformation that takes place.

A swing swings between two maximum heights, passing through the lowest point in the middle, as shown in the diagram.

1. a) At which point in the journey is its potential energy maximum?
2. b) Where is its kinetic energy maximum? Give reasons for both answers.

A block of $1\ \text{kg}$ falls from a height of $10\ \text{m}$ without friction (take $g = 10\ \text{m/s}^2$).

1. a) What potential energy does it have at the top?
2. b) If mechanical energy is conserved, what kinetic energy will it have when reaching the ground?

A ball of $2\ \text{kg}$ falls from a height of $5\ \text{m}$ without friction (take $g = 10\ \text{m/s}^2$).

1. a) How much potential energy does it lose during the fall?
2. b) How much kinetic energy does it gain?
3. c) At what speed does it reach the ground?

When braking a bicycle downhill, the brake pads get hot.

1. a) What has the kinetic energy of the bicycle been transformed into?
2. b) Has that energy been destroyed? Explain what the principle of conservation of energy states.

You throw a ball of $0.5\ \text{kg}$ upwards with a kinetic energy of $50\ \text{J}$ (take $g = 10\ \text{m/s}^2$). If there is no friction, what maximum height will it reach? Think about what energy the ball will have at the highest point.

In a water park there are two slides: one starts at a height of $3\ \text{m}$ and the other at $6\ \text{m}$, as shown in the diagram.

If there is no friction, on which slide will you reach the bottom with a higher speed? Explain why using energy concepts.

Name the main energy transformation that occurs in each case:

1. a) A switched-on battery-powered torch.
2. b) A solar panel charging a battery.
3. c) A hydroelectric power plant producing electricity.
4. d) You going for a run after breakfast.

A raindrop falls from a cloud at a height of $500\ \text{m}$ (take $g = 10\ \text{m/s}^2$).

1. a) If there were no air, at what speed would it reach the ground?
2. b) Does this result depend on the mass of the drop? Justify your answer.
3. c) Real raindrops reach the ground much more slowly: explain which force is involved and what the energy is transformed into.

A pendulum of $0.2\ \text{kg}$ oscillates and, at its highest point, has a potential energy of $40\ \text{J}$ relative to the lowest point. There is no friction. Look at the diagram:

1. a) What kinetic energy will it have when passing through the lowest point?
2. b) At what speed will it pass through there?

A girl of $30\ \text{kg}$ slides down from the top of a slide $7.2\ \text{m}$ high, without friction (take $g = 10\ \text{m/s}^2$).

1. a) What potential energy does she have at the top?
2. b) At what speed does she reach the bottom?

A roller coaster carriage of $200\ \text{kg}$ starts from the highest point, at $30\ \text{m}$, and passes through a dip at a height of $10\ \text{m}$ (take $g = 10\ \text{m/s}^2$, without friction).

1. a) How much potential energy has it lost?
2. b) How much kinetic energy has it gained? Explain which principle you used.

A ball of $0.4\ \text{kg}$ is dropped from $2\ \text{m}$, bounces on the ground and only rises back to $1.5\ \text{m}$ (take $g = 10\ \text{m/s}^2$), as shown in the diagram.

1. a) Has its mechanical energy been conserved?
2. b) How much energy was lost in the bounce?
3. c) What has that lost energy been transformed into?

A car of $800\ \text{kg}$ travelling at $20\ \text{m/s}$ brakes to a halt.

1. a) How much kinetic energy did it have?
2. b) Where has that energy gone?
3. c) Why is it dangerous to touch brake discs after a long downhill descent?

A skier glides down from the top of a slope, crosses the valley, and starts going up the opposite slope without pushing off with the poles, as shown in the diagram.

If there were no friction, what height on the opposite slope would they reach compared to the starting height? Justify your answer using the conservation of mechanical energy.

In a hydroelectric dam, $1000\ \text{kg}$ of water falls $50\ \text{m}$ onto the turbine (take $g = 10\ \text{m/s}^2$), as shown in the diagram.

1. a) How much potential energy does the water lose?
2. b) If the power plant used all that energy, how much electrical energy would it produce?
3. c) In reality it produces significantly less: explain why.

A package of $2\ \text{kg}$ falls from a stationary helicopter at a height of $45\ \text{m}$ (take $g = 10\ \text{m/s}^2$, without friction), as shown in the diagram.

1. a) What potential energy does the package have when it starts falling?
2. b) At what speed does it reach the ground?
3. c) When it has fallen halfway down, how much potential energy does it have left and how much kinetic energy does it have?

A child of $40\ \text{kg}$ goes down a slide $5\ \text{m}$ high (take $g = 10\ \text{m/s}^2$).

1. a) If there were no friction, at what speed would they reach the bottom?
2. b) In reality they reach $8\ \text{m/s}$: calculate their actual kinetic energy when reaching the bottom.
3. c) How much energy was lost due to friction? What has it been transformed into?

You release a pendulum from a certain height on the left and let it swing freely.

1. a) Describe how its kinetic and potential energies change over a full swing.
2. b) If there were no friction, to what height would it reach on the right side? Justify your answer.
3. c) In reality, each swing goes slightly less high: explain why and where the energy goes.

You throw a ball of $0.1\ \text{kg}$ vertically upwards at $20\ \text{m/s}$ (take $g = 10\ \text{m/s}^2$, without friction).

1. a) What kinetic energy does it have when leaving your hand?
2. b) What maximum height will it reach?
3. c) How much potential energy and how much kinetic energy does it have halfway up to that height?

In a skatepark there are two ramps starting at the same height: one is short and very steep, and the other is long and gentle. Two identical skateboarders release themselves at the same time, one on each ramp, without friction, as shown in the diagram.

1. a) Which of the two reaches the bottom at a higher speed? Give reasons using the conservation of energy.
2. b) Do they both reach the bottom at the same time? Explain the difference between final speed and descent time.

A roller coaster carriage of $100\ \text{kg}$ travels down from a peak of $30\ \text{m}$ and then tries to climb a hill of $25\ \text{m}$ (take $g = 10\ \text{m/s}^2$, without friction), as shown in the diagram.

1. a) Will it manage to get over the hill? Justify your answer by comparing potential energies.
2. b) At what speed will it pass over the crest of the hill?
3. c) What would happen if the hill were $35\ \text{m}$ high?

A toy rocket of $0.3\ \text{kg}$ takes off with a kinetic energy of $60\ \text{J}$ (take $g = 10\ \text{m/s}^2$).

1. a) If there were no air resistance, what maximum height would it reach?
2. b) In the actual launch it only goes up $15\ \text{m}$: how much energy has been dissipated?
3. c) What has the dissipated energy been transformed into?

In a classic demonstration, a teacher hangs a heavy ball from the ceiling, pulls it back until it touches their chin, and lets go without pushing it.

1. a) Explain, using the conservation of energy, why the ball does not hit their face when swinging back.
2. b) Taking air resistance into account, will it return to exactly the same height or a little lower?
3. c) What would happen if the teacher released it by pushing it forwards? Would it be dangerous? Justify your answer.