Kinetic Energy and Potential Energy
A bus climbing the Autopista Duarte, an elevated water tank full of water on top of a house, the spring on a door that closes by itself: three different objects that store mechanical energy in different ways. On this page you will learn the formula for each kind of energy, where it comes from, and how to solve for any quantity. The question that guides everything: why is a crash at 80 km/h much more than twice as dangerous as a crash at 40 km/h?
Translational kinetic energy
A shopping cart pushed gently does no harm. Thrown at full speed, it knocks over whatever is in its way. And a slow bus can do more damage than a fast motorcycle taxi. What does the energy of motion depend on?
Each factor has its reason. Mass enters directly: with twice the mass and the same speed there is twice the energy. Speed enters squared because, to bring an object up to a higher speed, the force acts over a longer distance and, at the same time, the object covers that distance faster. You will see this with the work-energy theorem.
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…
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: a) A passenger car of $1200\ \text{kg}$ travelling at $30\ \text{m/s}$. b) A lorry of $9000\ \text{kg}$ travelling at…
Calculate the kinetic energy of each vehicle:
- a) A passenger car of $1200\ \text{kg}$ travelling at $30\ \text{m/s}$.
- 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 $30\ \text{m/s}$. a) Calculate its kinetic energy in each case. b) How many times greater is the…
A car of $900\ \text{kg}$ travels first at $10\ \text{m/s}$ and then at $30\ \text{m/s}$.
- a) Calculate its kinetic energy in each case.
- b) How many times greater is the second than the first? Relate this factor to the factor by which the speed has been multiplied.
- c) Predict, using proportionality, the kinetic energy at $40\ \text{m/s}$ and check it using the formula.
- d) Draw a conclusion for road safety from the fact that kinetic energy increases with the square of the speed.
Rotational kinetic energy
A car tire that spins, the flywheel of an engine, or a dancer doing a merengue turn: their parts move in circles, even though the center does not move from its place. Do they have kinetic energy?
Yes. Each particle of the body has its own $\frac{1}{2} m_i v_i^2$. Since $v_i = \omega r_i$, where $\omega$ is the angular velocity in rad/s, adding up all the particles gives:
Notice the parallel with translation: $m \to I$ and $v \to \omega$.
| Body (axis through the center) | Moment of inertia $I$ |
|---|---|
| Thin hoop of radius $R$ | $M R^2$ |
| Solid disk or cylinder | $\frac{1}{2} M R^2$ |
| Solid sphere | $\frac{2}{5} M R^2$ |
| Rod of length $L$ | $\frac{1}{12} M L^2$ |
Gravitational potential energy
Setting a 5 kg weight down on your foot does not hurt. Dropping it from one meter up does. Where does the energy an object gains just by being raised come from?
The formula makes sense: to raise the object at constant speed you have to overcome its weight $mg$ over a distance $h$. The work $mg \cdot h$ is stored as energy.
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…
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.
An irrigation tank contains $200\ \text{kg}$ of water and its potential energy relative to the ground is $5880\ \text{J}$ (take $g = 9.8\ \text{m/s}^2$). a) Rearrange…
An irrigation tank contains $200\ \text{kg}$ of water and its potential energy relative to the ground is $5880\ \text{J}$ (take $g = 9.8\ \text{m/s}^2$).
- a) Rearrange for height in the formula $E_p = m \times g \times h$ and calculate the height of the water.
- b) If half of the tank is emptied, what will be the potential energy of the remaining water?
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…
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.
- a) Calculate its potential energy taking the surface of the table as the reference level.
- b) Calculate it taking the classroom floor as the reference level.
- c) Calculate it taking the street as the reference level.
- 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).
Elastic potential energy and Hooke's law
When you draw the string of a bow, the farther you pull it from its starting position, the more force you have to exert. How much energy is stored to launch the arrow?
A spring behaves this way: the force it exerts is proportional to its deformation. This is Hooke's law:
Since the force is not constant, we cannot simply use $F \cdot x$. The $F$–$x$ graph is a straight line starting at zero, and the work is the area under it: a triangle with base $x$ and height $kx$.
Try the simulator: change the force and watch the deformation.
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…
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.
- a) Calculate the stored elastic potential energy.
- 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 $2.25\ \text{J}$. a) Rearrange for the spring constant in…
The spring of a toy shock absorber is compressed by $0.15\ \text{m}$ and stores an elastic potential energy of $2.25\ \text{J}$.
- a) Rearrange for the spring constant in the formula $E_p = \frac{1}{2} \times k \times x^2$ and calculate it.
- 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…
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.
- a) What mistake has been made?
- b) Calculate the elastic potential energy correctly yourself.
- 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.
Mechanical energy: the sum
Indicate which mechanical energies each body possesses (kinetic, gravitational potential, or elastic potential; it may have more than one): a) A drawn bow with an arrow…
Indicate which mechanical energies each body possesses (kinetic, gravitational potential, or elastic potential; it may have more than one):
- a) A drawn bow with an arrow loaded, before being released.
- b) An aeroplane in cruise flight.
- c) A car parked at the top of a mountain pass.
- d) The spring of a freshly wound clockwork watch.
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…
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.
- a) Calculate its kinetic energy: does it change during the descent?
- b) Calculate the potential energy it loses.
- c) How does its mechanical energy change? Explain what has happened to that energy.
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…
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.
- a) Calculate its mechanical energy at the first point.
- b) Calculate its mechanical energy at the second point.
- c) By how much has its mechanical energy increased? Where has this energy come from?
- d) Is mechanical energy conserved during the rocket's ascent? Justify your answer.
Solving for quantities
Sometimes you are not asked for the energy, but for the speed, the mass, the height, or the spring constant. It is a matter of solving carefully.
Back to the beginning: since $E_c$ depends on $v^2$, a crash at 80 km/h releases four times the energy of one at 40 km/h. That is why limiting speed on the Autopista Duarte saves lives.
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 $E_c = \frac{1}{2} \times m \times v^2$…
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 cyclist and their bicycle have a combined mass of $80\ \text{kg}$ and descend a mountain pass at $15\ \text{m/s}$. a) Calculate their kinetic energy. b) A skater of…
A cyclist and their bicycle have a combined mass of $80\ \text{kg}$ and descend a mountain pass at $15\ \text{m/s}$.
- a) Calculate their kinetic energy.
- b) A skater of $45\ \text{kg}$ has exactly the same kinetic energy: rearrange for speed and calculate how fast she is travelling.
- 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. a) Rearrange for extension in the formula $E_p = \frac{1}{2} \times k \times x^2$…
A spring of $k = 800\ \text{N/m}$ stores $16\ \text{J}$ of elastic potential energy.
- a) Rearrange for extension in the formula $E_p = \frac{1}{2} \times k \times x^2$ and calculate it.
- b) What force does the spring exert in that situation? Use Hooke's law, $F = k \times x$.
- 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.