Energy Transformations and Conservation of Energy
You slide down a slide, drop a ball from the top of the bleachers, swing on a swing at the park. In every case, the energy you have up high gradually turns into motion as you come down. Nothing is created or destroyed: it only changes form. How can we use that idea to find out how fast you reach the bottom without measuring it?
Energy is transformed: potential ↔ kinetic
If you drop a steel marble from 5 cm onto a tray of modeling clay, it barely leaves a mark. But if you drop it from 2 m, it sinks in hard. Where does that speed come from if it started out at rest?
An apple detaches from a tree and falls to the ground, as shown in the diagram. a) What happens to its potential energy during the fall? b) And to its kinetic energy? c)…
An apple detaches from a tree and falls to the ground, as shown in the diagram.
- a) What happens to its potential energy during the fall?
- b) And to its kinetic energy?
- c) Explain the energy transformation that takes place.
A swing swings between two maximum heights, passing through the lowest point in the middle, as shown in the diagram. a) At which point in the journey is its potential…
A swing swings between two maximum heights, passing through the lowest point in the middle, as shown in the diagram.
- a) At which point in the journey is its potential energy maximum?
- b) Where is its kinetic energy maximum? Give reasons for both answers.
In a water park there are two slides: one starts at a height of $3\ \text{m}$ and the other at $6\ \text{m}$, as shown in the diagram. If there is no friction, on which…
In a water park there are two slides: one starts at a height of $3\ \text{m}$ and the other at $6\ \text{m}$, as shown in the diagram.
If there is no friction, on which slide will you reach the bottom with a higher speed? Explain why using energy concepts.
Conservation of mechanical energy (without friction)
If you use a photogate to measure the speed of a cart rolling down from a height of 1 m without friction, it always reads about 4.5 m/s, whether the ramp is straight or has curves. Why does the final speed depend only on the drop in height and not on the path?
We use $E_p = m g h$ and $E_c = \frac12 m v^2$, with $g \approx 10\ \text{m/s}^2$ in these exercises (the more precise value is $9.8\ \text{m/s}^2$).
Notice that in $m g h = \frac12 m v^2$ the mass cancels, so $v = \sqrt{2 g h}$. That is why the final speed depends only on the height, not on the shape of the ramp.
A block of $1\ \text{kg}$ falls from a height of $10\ \text{m}$ without friction (take $g = 10\ \text{m/s}^2$). a) What potential energy does it have at the top? b) If…
A block of $1\ \text{kg}$ falls from a height of $10\ \text{m}$ without friction (take $g = 10\ \text{m/s}^2$).
- a) What potential energy does it have at the top?
- b) If mechanical energy is conserved, what kinetic energy will it have when reaching the ground?
A ball of $2\ \text{kg}$ falls from a height of $5\ \text{m}$ without friction (take $g = 10\ \text{m/s}^2$). a) How much potential energy does it lose during the fall?…
A ball of $2\ \text{kg}$ falls from a height of $5\ \text{m}$ without friction (take $g = 10\ \text{m/s}^2$).
- a) How much potential energy does it lose during the fall?
- b) How much kinetic energy does it gain?
- c) At what speed does it reach the ground?
A girl of $30\ \text{kg}$ slides down from the top of a slide $7.2\ \text{m}$ high, without friction (take $g = 10\ \text{m/s}^2$). a) What potential energy does she…
A girl of $30\ \text{kg}$ slides down from the top of a slide $7.2\ \text{m}$ high, without friction (take $g = 10\ \text{m/s}^2$).
- a) What potential energy does she have at the top?
- b) At what speed does she reach the bottom?
When there is friction: energy doesn't disappear, it is transformed
A rubber ball bounces less and less high until it stops, and if you touch it, it is a little warm. Where did the mechanical energy it had at the beginning go?
When braking a bicycle downhill, the brake pads get hot. a) What has the kinetic energy of the bicycle been transformed into? b) Has that energy been destroyed? Explain…
When braking a bicycle downhill, the brake pads get hot.
- a) What has the kinetic energy of the bicycle been transformed into?
- b) Has that energy been destroyed? Explain what the principle of conservation of energy states.
A ball of $0.4\ \text{kg}$ is dropped from $2\ \text{m}$, bounces on the ground and only rises back to $1.5\ \text{m}$ (take $g = 10\ \text{m/s}^2$), as shown in the…
A ball of $0.4\ \text{kg}$ is dropped from $2\ \text{m}$, bounces on the ground and only rises back to $1.5\ \text{m}$ (take $g = 10\ \text{m/s}^2$), as shown in the diagram.
- a) Has its mechanical energy been conserved?
- b) How much energy was lost in the bounce?
- c) What has that lost energy been transformed into?
A car of $800\ \text{kg}$ travelling at $20\ \text{m/s}$ brakes to a halt. a) How much kinetic energy did it have? b) Where has that energy gone? c) Why is it dangerous…
A car of $800\ \text{kg}$ travelling at $20\ \text{m/s}$ brakes to a halt.
- a) How much kinetic energy did it have?
- b) Where has that energy gone?
- c) Why is it dangerous to touch brake discs after a long downhill descent?
Problems using conservation of energy
A roller coaster carriage of $200\ \text{kg}$ starts from the highest point, at $30\ \text{m}$, and passes through a dip at a height of $10\ \text{m}$ (take…
A roller coaster carriage of $200\ \text{kg}$ starts from the highest point, at $30\ \text{m}$, and passes through a dip at a height of $10\ \text{m}$ (take $g = 10\ \text{m/s}^2$, without friction).
- a) How much potential energy has it lost?
- b) How much kinetic energy has it gained? Explain which principle you used.
A package of $2\ \text{kg}$ falls from a stationary helicopter at a height of $45\ \text{m}$ (take $g = 10\ \text{m/s}^2$, without friction), as shown in the diagram. a)…
A package of $2\ \text{kg}$ falls from a stationary helicopter at a height of $45\ \text{m}$ (take $g = 10\ \text{m/s}^2$, without friction), as shown in the diagram.
- a) What potential energy does the package have when it starts falling?
- b) At what speed does it reach the ground?
- c) When it has fallen halfway down, how much potential energy does it have left and how much kinetic energy does it have?
A roller coaster carriage of $100\ \text{kg}$ travels down from a peak of $30\ \text{m}$ and then tries to climb a hill of $25\ \text{m}$ (take $g = 10\ \text{m/s}^2$,…
A roller coaster carriage of $100\ \text{kg}$ travels down from a peak of $30\ \text{m}$ and then tries to climb a hill of $25\ \text{m}$ (take $g = 10\ \text{m/s}^2$, without friction), as shown in the diagram.
- a) Will it manage to get over the hill? Justify your answer by comparing potential energies.
- b) At what speed will it pass over the crest of the hill?
- c) What would happen if the hill were $35\ \text{m}$ high?
Applications: from the mountain to the hydroelectric power plant
In Mexico, dams such as Chicoasén, in Chiapas, produce electricity with water that falls from a great height. How much energy can the water provide, and why is it never all used?
Name the main energy transformation that occurs in each case: a) A switched-on battery-powered torch. b) A solar panel charging a battery. c) A hydroelectric power plant…
Name the main energy transformation that occurs in each case:
- a) A switched-on battery-powered torch.
- b) A solar panel charging a battery.
- c) A hydroelectric power plant producing electricity.
- d) You going for a run after breakfast.
In a hydroelectric dam, $1000\ \text{kg}$ of water falls $50\ \text{m}$ onto the turbine (take $g = 10\ \text{m/s}^2$), as shown in the diagram. a) How much potential…
In a hydroelectric dam, $1000\ \text{kg}$ of water falls $50\ \text{m}$ onto the turbine (take $g = 10\ \text{m/s}^2$), as shown in the diagram.
- a) How much potential energy does the water lose?
- b) If the power plant used all that energy, how much electrical energy would it produce?
- c) In reality it produces significantly less: explain why.
A toy rocket of $0.3\ \text{kg}$ takes off with a kinetic energy of $60\ \text{J}$ (take $g = 10\ \text{m/s}^2$). a) If there were no air resistance, what maximum height…
A toy rocket of $0.3\ \text{kg}$ takes off with a kinetic energy of $60\ \text{J}$ (take $g = 10\ \text{m/s}^2$).
- a) If there were no air resistance, what maximum height would it reach?
- b) In the actual launch it only goes up $15\ \text{m}$: how much energy has been dissipated?
- c) What has the dissipated energy been transformed into?
Back to the beginning: the speed at the bottom of a slide or a fall depends only on the height you lose, because potential energy is converted into kinetic energy and, if there is friction, part of it goes into heat.