Exercises: Mechanical energy: kinetic and potential

Kinetic energy is calculated as $E_c = \frac{1}{2} \times m \times v^2$. A toy car of $2\ \text{kg}$ moves at $3\ \text{m/s}$. Calculate its kinetic energy.

Potential energy is calculated as $E_p = m \times g \times h$. A book of $1\ \text{kg}$ is on a shelf $5\ \text{m}$ off the ground (take $g = 10\ \text{m/s}^2$), as shown in the diagram.

Calculate its potential energy.

State what type of mechanical energy each body has and explain why:

1. a) A car driving along a motorway.
2. b) A rock on the edge of a cliff.
3. c) An aeroplane in mid-flight.

In what unit is energy measured in the International System? What other physical quantity shares this same unit? Explain why it makes sense that they share it.

A soccer ball of $0.5\ \text{kg}$ is kicked off at $20\ \text{m/s}$. What kinetic energy does it have?

A plant pot of $2\ \text{kg}$ is on a balcony railing, $1.5\ \text{m}$ above the ground (take $g = 9.8\ \text{m/s}^2$). Calculate its potential energy.

Complete the sentence with the missing words: "The kinetic energy of a body depends on its $\dots$ and its $\dots$; potential energy depends on its $\dots$ and the $\dots$ at which it is located."

Two cyclists are travelling at the same speed, but one is carrying a rucksack loaded with books and their total mass is greater. Which of the two has more kinetic energy? Reason your answer without doing calculations.

A cyclist and their bicycle have a total mass of $60\ \text{kg}$ and climb a mountain pass $200\ \text{m}$ high (take $g = 10\ \text{m/s}^2$). How much potential energy have they gained upon reaching the top?

A ball of $0.2\ \text{kg}$ rolls first at $5\ \text{m/s}$ and then at $10\ \text{m/s}$.

1. a) Calculate its kinetic energy in each case.
2. b) How many times greater is the second than the first? Explain why it does not double when the speed is doubled.

A car of $1000\ \text{kg}$ travels at $72\ \text{km/h}$.

1. a) Express its speed in $\text{m/s}$.
2. b) Calculate its kinetic energy.

A painting of $4\ \text{kg}$ is hung first $2\ \text{m}$ off the floor and then raised to $6\ \text{m}$ (take $g = 10\ \text{m/s}^2$), as shown in the diagram.

1. a) Calculate its potential energy in each position.
2. b) What is the relationship between height and potential energy? Express it using the numbers above.

Which of these two bodies has more mechanical energy (kinetic plus potential)? Calculate it for each one (take $g = 10\ \text{m/s}^2$):

1. a) A crane of $5\ \text{kg}$ flying at $10\ \text{m/s}$ at a height of $100\ \text{m}$.
2. b) A stone of $10\ \text{kg}$ at rest at a height of $300\ \text{m}$.

The diagram summarises the situation:

A trolley of $2\ \text{kg}$ has a kinetic energy of $100\ \text{J}$. At what speed is it moving? First rearrange the kinetic energy formula for speed.

A child of $30\ \text{kg}$ is at the top of a slide and their potential energy is $600\ \text{J}$ (take $g = 10\ \text{m/s}^2$). How high is the slide's platform?

A remote-controlled car travels at $4\ \text{m/s}$ and its kinetic energy is $40\ \text{J}$.

1. a) What is its mass?
2. b) What kinetic energy would it have if it travelled at $8\ \text{m/s}$?

A passenger car of $1000\ \text{kg}$ travels at $30\ \text{m/s}$ and a lorry of $4000\ \text{kg}$ travels at $15\ \text{m/s}$.

1. a) Calculate the kinetic energy of each vehicle.
2. b) Which one has more kinetic energy? Did you expect that result?
3. c) Explain, using the formula, why speed "weighs" more than mass in kinetic energy.

A drone of $2\ \text{kg}$ flies at a height of $20\ \text{m}$ with a speed of $10\ \text{m/s}$ (take $g = 10\ \text{m/s}^2$), as shown in the diagram.

1. a) Calculate its kinetic energy.
2. b) Calculate its potential energy.
3. c) What is its total mechanical energy?
4. d) If it stops in mid-air while maintaining its height, what happens to each of those energies?

Your friend says: "If I run twice as fast, I will have twice as much kinetic energy."

1. a) Calculate the kinetic energy of a person of $50\ \text{kg}$ running at $5\ \text{m/s}$ and at $10\ \text{m/s}$.
2. b) Was your friend right? If not, correct their statement.
3. c) What consequence does this have for road safety when a car doubles its speed?

A rock of $20\ \text{kg}$ is placed at a height of $10\ \text{m}$, first on Earth ($g = 10\ \text{m/s}^2$) and then on the Moon ($g = 1.6\ \text{m/s}^2$), as shown in the diagram.

1. a) Calculate its potential energy in each location.
2. b) In which of the two locations would the falling rock be more dangerous? Justify your answer using energy values.
3. c) Has the mass of the rock changed when taken to the Moon?

Order from highest to lowest kinetic energy, calculating it first in each case:

1. a) A bullet of $0.02\ \text{kg}$ at $400\ \text{m/s}$.
2. b) An athlete of $80\ \text{kg}$ at $10\ \text{m/s}$.
3. c) A bicycle of $10\ \text{kg}$ at $20\ \text{m/s}$.

Comment on what surprises you about the result.

You have a steel ball, a wooden ball of the same size, and a shelving unit with shelves at different heights.

1. a) Explain how you would test that potential energy depends on mass, stating what you would compare.
2. b) How would you test that it depends on height?
3. c) What quantity must you measure besides height, and how would you use it in the formula $E_p = m \times g \times h$?

A lift of $400\ \text{kg}$ ascends $12\ \text{m}$ in $8\ \text{s}$ (take $g = 10\ \text{m/s}^2$).

1. a) How much potential energy does the lift car gain?
2. b) What power does the motor develop?
3. c) Relate the work done by the motor to the potential energy gained by the lift car.

An advert says: "Our electric car reaches twice the speed of the previous model while requiring only twice as much energy."

1. a) Calculate the kinetic energy of a car of $800\ \text{kg}$ at $10\ \text{m/s}$ and at $20\ \text{m/s}$.
2. b) How many times more energy is needed to travel at twice the speed?
3. c) Evaluate whether the advert's claim is correct and explain its mistake.