Kinetic Energy and Potential Energy

Kinetic energy is calculated as $E_c = \frac{1}{2} \times m \times v^2$. A volleyball of $0.3\ \text{kg}$ reaches the player's hands at $24\ \text{m/s}$. Calculate its kinetic energy.

A boy and his scooter have a combined mass of $60\ \text{kg}$ and travel at $18\ \text{km/h}$.

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

Your classmate calculates the kinetic energy of a car of $1000\ \text{kg}$ travelling at $90\ \text{km/h}$ by substituting $v = 90$ into the formula, and obtains $4050000\ \text{J}$.

1. a) What mistake has your classmate made?
2. b) Calculate the kinetic energy correctly yourself, first converting the speed to $\text{m/s}$.
3. c) Explain why it is compulsory to always work in SI units.

Potential energy is calculated as $E_p = m \times g \times h$. A lamp of $3\ \text{kg}$ hangs from the ceiling $2.5\ \text{m}$ above the floor (take $g = 9.8\ \text{m/s}^2$). Calculate its potential energy.

A rooftop water tank contains $500\ \text{kg}$ of water $15\ \text{m}$ above the ground (take $g = 9.8\ \text{m/s}^2$).

1. a) Calculate its potential energy, in joules.
2. b) Express it in kilojoules.
3. c) If the building were $3\ \text{m}$ taller, how much additional potential energy would the water have?

An astronaut lifts a rock of $20\ \text{kg}$ to a height of $1.5\ \text{m}$, first on Earth ($g = 9.8\ \text{m/s}^2$) and then on Mars ($g = 3.7\ \text{m/s}^2$).

1. a) Calculate the potential energy gained by the rock on each planet.
2. b) Where is it "harder" to lift it? Justify your answer using your results.
3. c) Has the mass of the rock changed upon travelling to Mars? What about its weight?

State what type of mechanical energy each object has (kinetic, potential, or both) and how it is changing:

1. a) A funicular railway ascending at a constant speed.
2. b) A diver standing still on the diving board before jumping.
3. c) A rocket accelerating upwards after launch.

A toy helicopter of $0.8\ \text{kg}$ flies at $3\ \text{m/s}$ at a height of $10\ \text{m}$ (take $g = 10\ \text{m/s}^2$), as shown in the figure.

1. a) Calculate its kinetic energy.
2. b) Calculate its potential energy.
3. c) What is its total mechanical energy?

Complete the sentence with the missing words: "The mechanical energy of an object is the $\dots$ of its $\dots$ energy and its $\dots$ energy". Then give an example of an object that has both types of energy at the same time.

A peregrine falcon of $1\ \text{kg}$ dives at $60\ \text{m/s}$ when it is at a height of $500\ \text{m}$, while a pigeon of $0.4\ \text{kg}$ flies at $20\ \text{m/s}$ at a height of $100\ \text{m}$ (take $g = 10\ \text{m/s}^2$), as shown in the figure.

1. a) Calculate the kinetic energy and potential energy of each bird.
2. b) Which bird has more total mechanical energy?

A microlight aircraft of $300\ \text{kg}$ flies at a height of $500\ \text{m}$ with a speed of $40\ \text{m/s}$ (take $g = 10\ \text{m/s}^2$).

1. a) Calculate its kinetic energy.
2. b) Calculate its potential energy.
3. c) What is its total mechanical energy?
4. d) If it descends to $200\ \text{m}$ while maintaining the same speed, how much mechanical energy does it lose? Where do you think that energy goes?

A skier of $80\ \text{kg}$ travels down a slope at $12\ \text{m/s}$. What is his kinetic energy?

A cyclist and her bicycle have a combined mass of $90\ \text{kg}$ and travel at $36\ \text{km/h}$.

1. a) Express the speed in $\text{m/s}$.
2. b) Calculate the kinetic energy.
3. c) After stopping pedalling and braking gently, her speed drops to $18\ \text{km/h}$: calculate the new kinetic energy and the energy lost.

Calculate the kinetic energy of each of the following and determine which has more:

1. a) A rugby player of $100\ \text{kg}$ running at $8\ \text{m/s}$ during a counter-attack.
2. b) A golf ball of $0.045\ \text{kg}$ leaving the tee at $90\ \text{m/s}$.

Comment on what strikes you as surprising or interesting about the result.

A high-speed train of $300\ \text{t}$ travels at $180\ \text{km/h}$.

1. a) Express its mass in kilograms and its speed in $\text{m/s}$.
2. b) Calculate its kinetic energy and express it in standard form.
3. c) A passenger car of $1200\ \text{kg}$ travels at $25\ \text{m/s}$: calculate its kinetic energy and determine how many times greater the train's kinetic energy is.

A car of $1200\ \text{kg}$ travels at $20\ \text{m/s}$.

1. a) Calculate its kinetic energy.
2. b) A motorbike and its rider have a combined mass of $300\ \text{kg}$ and have the same kinetic energy: at what speed is it travelling?
3. c) Explain, using the formula, why having a quarter of the mass corresponds to exactly double the speed.

You want to verify in the laboratory that kinetic energy is proportional to the square of speed, using a trolley of $4\ \text{kg}$, a ramp, and a speed sensor, as shown in the figure.

1. a) Explain what you would measure and how you would organise the data.
2. b) Predict the results: if at $1\ \text{m/s}$ the kinetic energy is $2\ \text{J}$, what will it be at $2\ \text{m/s}$ and at $3\ \text{m/s}$?
3. c) If you plotted kinetic energy against speed, what shape would the graph have? What if you plotted it against $v^2$?

A bag hanging from a peg, $2\ \text{m}$ above the ground, has a potential energy of $140\ \text{J}$ (take $g = 10\ \text{m/s}^2$). What is the mass of the bag? Rearrange the potential energy formula to solve for mass first.

A javelin of $0.8\ \text{kg}$ has a kinetic energy of $90\ \text{J}$. Rearrange the formula $E_c = \frac{1}{2} \times m \times v^2$ to solve for speed and calculate it.

A cat of $4\ \text{kg}$ is sitting on top of a wall and its potential energy relative to the ground is $196\ \text{J}$ (take $g = 9.8\ \text{m/s}^2$). How high is the wall?

A lorry travels at $15\ \text{m/s}$ and has a kinetic energy of $360000\ \text{J}$. What is its total mass? Rearrange the kinetic energy formula to solve for mass first.

A squirrel of $0.3\ \text{kg}$ is on a branch $4\ \text{m}$ above the ground and then climbs down the trunk to its nest, at $2.5\ \text{m}$ (take $g = 10\ \text{m/s}^2$), as shown in the figure.

1. a) Calculate its potential energy in each location.
2. b) How much potential energy did it lose by climbing down to the nest?

A soccer ball of $0.45\ \text{kg}$ bounces off at $40\ \text{m/s}$.

1. a) Calculate its kinetic energy.
2. b) At what speed would an athlete of $80\ \text{kg}$ have to run to have the same kinetic energy? Rearrange the formula for speed first.

A rescue helicopter of $3000\ \text{kg}$ ascends from a height of $100\ \text{m}$ to $300\ \text{m}$ while maintaining a constant speed of $20\ \text{m/s}$ (take $g = 10\ \text{m/s}^2$).

1. a) Calculate its kinetic energy: does it change during the ascent?
2. b) Calculate the potential energy it gains.
3. c) How does its mechanical energy change? Where does this energy come from?