Kinetic energy and potential energy

Kinetic energy is calculated as $E_c = \frac{1}{2} \cdot m \cdot v^2$. A handball goalkeeper throws the ball, of mass $0,4\ \text{kg}$, at $25\ \text{m/s}$. Calculate its kinetic energy.

Calculate the 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.

Gravitational potential energy is calculated as $E_p = m \cdot g \cdot h$. 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 $E_p = m \cdot g \cdot h$ and calculate it.

An irrigation tank contains $200\ \text{kg}$ of water and its potential energy relative to the ground is $5880\ \text{J}$ (take $g = 9,8\ \text{m/s}^2$).

1. a) Rearrange for height in the formula $E_p = m \cdot g \cdot h$ and calculate the height of the water.
2. b) If half of the tank is emptied, what will be the potential energy of the remaining water?

The elastic potential energy of a spring is calculated as $E_p = \frac{1}{2} \cdot k \cdot x^2$, 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.

The spring of a toy shock absorber is compressed by $0,15\ \text{m}$ and stores an elastic potential energy of $2,25\ \text{J}$.

1. a) Rearrange for the spring constant in the formula $E_p = \frac{1}{2} \cdot k \cdot x^2$ and calculate it.
2. b) What energy would it store if compressed by $0,3\ \text{m}$?

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 $E_p = k \cdot x^2$. 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 $k \cdot x$, 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.

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 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.

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, $4/3$.

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

1. a) Rearrange for extension in the formula $E_p = \frac{1}{2} \cdot k \cdot x^2$ and calculate it.
2. b) What force does the spring exert in that situation? Use Hooke's law, $F = k \cdot x$.
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.