Kinetic Energy and Potential Energy

Kinetic energy is calculated as $E_c = \frac{1}{2} \times m \times v^2$. A handball goalkeeper throws the ball, of mass $0.4\ \text{kg}$, at $25\ \text{m/s}$. Calculate its kinetic energy.

Calculate the kinetic energy of each vehicle:

1. a) A passenger car of $1200\ \text{kg}$ travelling at $30\ \text{m/s}$.
2. b) A lorry of $9000\ \text{kg}$ travelling at $12\ \text{m/s}$.

Which one has more kinetic energy? Comment on which factor has a greater impact on the result: mass or speed.

A car of $900\ \text{kg}$ travels first at $10\ \text{m/s}$ and then at $30\ \text{m/s}$.

1. a) Calculate its kinetic energy in each case.
2. b) How many times greater is the second than the first? Relate this factor to the factor by which the speed has been multiplied.
3. c) Predict, using proportionality, the kinetic energy at $40\ \text{m/s}$ and check it using the formula.
4. d) Draw a conclusion for road safety from the fact that kinetic energy increases with the square of the speed.

Gravitational potential energy is calculated as $E_p = m \times g \times h$. A flower pot of $6\ \text{kg}$ is on a window sill, $4\ \text{m}$ above the ground (take $g = 10\ \text{m/s}^2$). Calculate its potential energy relative to the ground.

A light bulb hanging from the ceiling, $2.5\ \text{m}$ above the ground, has a potential energy of $98\ \text{J}$ (take $g = 9.8\ \text{m/s}^2$). Rearrange for mass in the formula $E_p = m \times g \times h$ and calculate it.

A book of $0.5\ \text{kg}$ is on a table $80\ \text{cm}$ high, in a second-floor classroom located $8\ \text{m}$ above the street (take $g = 10\ \text{m/s}^2$). The figure shows the three levels.

1. a) Calculate its potential energy taking the surface of the table as the reference level.
2. b) Calculate it taking the classroom floor as the reference level.
3. c) Calculate it taking the street as the reference level.
4. d) Explain what it means to choose a reference level and explain why, in energy conservation problems, the result does not depend on the chosen reference level (think about which quantity actually appears: height or difference in height).

The elastic potential energy of a spring is calculated as $E_p = \frac{1}{2} \times k \times x^2$, where $k$ is the spring constant and $x$ is the extension. A spring with $k = 300\ \text{N/m}$ is stretched $0.1\ \text{m}$. The diagram is shown in the figure.

1. a) Calculate the stored elastic potential energy.
2. b) If instead of being stretched it is compressed by $0.1\ \text{m}$, is the stored energy greater, less, or equal? Justify your answer by looking at how $x$ appears in the formula.

Two identical springs are stretched in the laboratory: the first by $5\ \text{cm}$ and the second by $10\ \text{cm}$. Without performing numerical calculations, use the formula $E_p = \frac{1}{2} \times k \times x^2$ to reason which one stores more elastic energy and how many times more.

A spring with constant $k = 300\ \text{N/m}$ is stretched in stages in the laboratory.

1. a) Calculate the elastic potential energy when $x = 0.1\ \text{m}$.
2. b) Calculate the energy when $x = 0.2\ \text{m}$.
3. c) How many times greater is the second than the first? Explain the result from the formula and predict, without using a calculator, the energy for $x = 0.4\ \text{m}$.

Your classmate calculates the elastic potential energy of a $k = 500\ \text{N/m}$ spring compressed by $0.2\ \text{m}$ and gets $20\ \text{J}$, because he used the expression $E_p = k \times x^2$. The figure shows how the required force grows as the spring is compressed.

1. a) What mistake has been made?
2. b) Calculate the elastic potential energy correctly yourself.
3. c) The factor $\frac{1}{2}$ appears because, when stretching or compressing the spring, the required force increases from $0$ to $k \times x$, and the work done is calculated using the average force. Calculate the average force in this case and check that average force multiplied by extension gives the correct result.

Indicate which mechanical energies each body possesses (kinetic, gravitational potential, or elastic potential; it may have more than one):

1. a) A drawn bow with an arrow loaded, before being released.
2. b) An aeroplane in cruise flight.
3. c) A car parked at the top of a mountain pass.
4. d) The spring of a freshly wound clockwork watch.

A delivery drone of $2\ \text{kg}$ flies at $3\ \text{m/s}$ at a height of $30\ \text{m}$ (take $g = 10\ \text{m/s}^2$).

1. a) Calculate its kinetic energy.
2. b) Calculate its potential energy.
3. c) What is its mechanical energy, $E_m = E_c + E_p$?

A motorbike with its rider has a total mass of $250\ \text{kg}$ and travels at $72\ \text{km/h}$.

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

A delivery drone of $3\ \text{kg}$ descends from a height of $50\ \text{m}$ to $20\ \text{m}$ while maintaining a constant speed of $8\ \text{m/s}$ (take $g = 10\ \text{m/s}^2$). The situation is shown in the figure.

1. a) Calculate its kinetic energy: does it change during the descent?
2. b) Calculate the potential energy it loses.
3. c) How does its mechanical energy change? Explain what has happened to that energy.

A high-speed train of $400\ \text{t}$ travels at $180\ \text{km/h}$.

1. a) Express its mass in kilograms, in standard form, and its speed in $\text{m/s}$.
2. b) Calculate its kinetic energy and express it in standard form.
3. c) Calculate the kinetic energy it would have travelling at $90\ \text{km/h}$ and determine how it relates to the previous value.

Calculate the mechanical energies of these two vehicles and compare them (take $g = 9.8\ \text{m/s}^2$):

1. a) A commercial airliner of $80000\ \text{kg}$ flying at a height of $10000\ \text{m}$ at $250\ \text{m/s}$: calculate its kinetic energy, its potential energy and its mechanical energy, in standard form.
2. b) A Formula 1 car of $800\ \text{kg}$ racing at $80\ \text{m/s}$ on a circuit at sea level: calculate its kinetic energy and explain why its potential energy can be taken as zero.
3. c) Comment on which energy term dominates in each case and why.

A sounding rocket of $500\ \text{kg}$ ascends while accelerating. When it is at a height of $1000\ \text{m}$, its speed is $100\ \text{m/s}$, and a few seconds later it passes a height of $3000\ \text{m}$ at $200\ \text{m/s}$ (take $g = 10\ \text{m/s}^2$). The figure shows the two points of the ascent.

1. a) Calculate its mechanical energy at the first point.
2. b) Calculate its mechanical energy at the second point.
3. c) By how much has its mechanical energy increased? Where has this energy come from?
4. d) Is mechanical energy conserved during the rocket's ascent? Justify your answer.

A golf ball of mass $0.045\ \text{kg}$ leaves the tee with a kinetic energy of $81\ \text{J}$. Rearrange for speed in the formula $E_c = \frac{1}{2} \times m \times v^2$ and calculate the speed with which it leaves the tee.

A person riding an electric scooter travels at $10\ \text{m/s}$ and their kinetic energy is $3600\ \text{J}$. Rearrange for mass in the kinetic energy formula and calculate the total mass of the person-scooter combined system.

A cyclist and their bicycle have a combined mass of $80\ \text{kg}$ and descend a mountain pass at $15\ \text{m/s}$.

1. a) Calculate their kinetic energy.
2. b) A skater of $45\ \text{kg}$ has exactly the same kinetic energy: rearrange for speed and calculate how fast she is travelling.
3. c) The ratio of masses is $80/45 = 16/9$. Explain, using the kinetic energy formula, why the ratio of speeds is the square root of the inverted ratio of masses, that is, $4/3$.

You want to verify in the laboratory that elastic potential energy is proportional to the square of the extension, using a spring of $k = 200\ \text{N/m}$, a ruler, and a set of weights.

1. a) Explain what you would measure and how you would obtain the stored energy for each measurement.
2. b) Predict the results: calculate the potential energy for extensions of $0.05\ \text{m}$, $0.10\ \text{m}$, and $0.15\ \text{m}$.
3. c) If you plotted energy against extension $x$, what shape would the graph have? What if you plotted it against $x^2$? Justify your answer.

A spring of $k = 800\ \text{N/m}$ stores $16\ \text{J}$ of elastic potential energy.

1. a) Rearrange for extension in the formula $E_p = \frac{1}{2} \times k \times x^2$ and calculate it.
2. b) What force does the spring exert in that situation? Use Hooke's law, $F = k \times x$.
3. c) What average force was required to bring it from its natural position to that extension? Check that the work done by this average force equals the stored energy.