Mechanical Energy and Conservation of Energy, Work, Energy, and Simple Machines | Physics

Learning objectives

Teacher Support

The learning objectives in this section will help your students master the following standards:

(6) Science concepts. The student knows that changes occur within a physical system and applies the laws of conservation of energy and momentum. The student is expected to:
- (B) investigate examples of kinetic and potential energy and their transformations;
- (D) demonstrate and apply the laws of conservation of energy and conservation of momentum in one dimension.

In addition, the High School Physics Laboratory Manual addresses content in this section in the lab titled: Work and Energy, as well as the following standards:

(6) Science concepts. The student knows that changes occur within a physical system and applies the laws of conservation of energy and momentum. The student is expected to:
- (B) investigate examples of kinetic and potential energy and their transformations;
- (D) demonstrate and apply the laws of conservation of energy and conservation of momentum in one dimension.

Section Key Terms

law of conservation of energy

Teacher Support

[BL][OL] Begin by distinguishing mechanical energy from other forms of energy. Explain how the general definition of energy as the ability to do work makes perfect sense in terms of either form of mechanical energy. Discuss the law of conservation of energy and dispel any misconceptions related to this law, such is the idea that moving objects just slow down naturally. Identify heat generated by friction as the usual explanation for apparent violations of the law.

[AL] Start a discussion about how other useful forms of energy also end up as wasted heat, such as light, sound, and electricity. Try to get students to understand heat and temperature at a molecular level. Explain that energy lost to friction is really transforming kinetic energy at the macroscopic level to kinetic energy at the atomic level.

Mechanical Energy and Conservation of Energy

We saw earlier that mechanical energy can be either potential or kinetic. In this section we will see how energy is transformed from one of these forms to the other. We will also see that, in a closed system, the sum of these forms of energy remains constant.

Quite a bit of potential energy is gained by a roller coaster car and its passengers when they are raised to the top of the first hill. Remember that the potential part of the term means that energy has been stored and can be used at another time. You will see that this stored energy can either be used to do work or can be transformed into kinetic energy. For example, when an object that has gravitational potential energy falls, its energy is converted to kinetic energy. Remember that both work and energy are expressed in joules.

Refer back to Figure_09_01_Lifting. The amount of work required to raise the TV from point A to point B is equal to the amount of gravitational potential energy the TV gains from its height above the ground. This is generally true for any object raised above the ground. If all the work done on an object is used to raise the object above the ground, the amount of work equals the object’s gain in gravitational potential energy. However, note that because of the work done by friction, these energy–work transformations are never perfect. Friction causes the loss of some useful energy. In the discussions to follow, we will use the approximation that transformations are frictionless.

Now, let’s look at the roller coaster in Figure_09_02_roller. Work was done on the roller coaster to get it to the top of the first rise; at this point, the roller coaster has gravitational potential energy. It is moving slowly, so it also has a small amount of kinetic energy. As the car descends the first slope, its PE is converted to KE. At the low point much of the original PE has been transformed to KE, and speed is at a maximum. As the car moves up the next slope, some of the KE is transformed back into PE and the car slows down.

View original figure (external; image not reproduced in this text edition)

Original figure description: An illustration shows a side view of a roller coaster. People in a cart are at the top of a hill, and the cart is labeled Start. The cart is heading towards a downward slope. The opposite end of the roller coaster is flat and is labeled Finish.

During this roller coaster ride, there are conversions between potential and kinetic energy.

Teacher Support

[OL][AL] Ask if definitions of energy make sense to the class, and try to bring out any expressions of confusions or misconceptions. Help them make the logical leap that, if energy is the ability to do work, it makes sense that it is expressed by the same unit of measurement. Ask students to name all the forms of energy they can. Ask if this helps them get a feel for the nature of energy. Ask if they have a problem seeing how some forms of energy, such as sunlight, can do work.

[BL][OL] You may want to introduce the concept of a reference point as the starting point of motion. Relate this to the origin of a coordinate grid.

[BL] Make it clear that energy is a different property with different units than either force or power.

[OL] Help students understand that the speed with which the TV is delivered is not part of the calculation of PE. It is assumed that the speed is constant. Any KE due to increases in delivery speed will be lost when motion stops.

[BL] Be sure there is a clear understanding of the distinction between kinetic and potential energy and between velocity and acceleration. Explain that the word potential means that the energy is available but it does not mean that it has to be used or will be used.

On an actual roller coaster, there are many ups and downs, and each of these is accompanied by transitions between kinetic and potential energy. Assume that no energy is lost to friction. At any point in the ride, the total mechanical energy is the same, and it is equal to the energy the car had at the top of the first rise. This is a result of the law of conservation of energy, which says that, in a closed system, total energy is conservedthat is, it is constant. Using subscripts 1 and 2 to represent initial and final energy, this law is expressed as

K E_{1} + P E_{1} = K E_{2} + P E_{2} .

Either side equals the total mechanical energy. The phrase in a closed system means we are assuming no energy is lost to the surroundings due to friction and air resistance. If we are making calculations on dense falling objects, this is a good assumption. For the roller coaster, this assumption introduces some inaccuracy to the calculation.

Calculations involving Mechanical Energy and Conservation of Energy

Teacher Support

[BL][OL] Impress upon the students the significant amount of work required to get a roller coaster car to the top of the first, highest point. Compare it to the amount of work it would take to walk to the top of the roller coaster. Ask students why they may feel tired if they had to walk or climb to the top of the roller coaster (they have to use energy to exert the force required to move their bodies upwards against the force of gravity). Check if students can correctly predict that the ratio of the mass of the car to a person’s mass would be the ratio of work done and energy gained (for example, if the car’s mass was 10 times a person’s mass, the amount of work needed to move the car to the top of the hill would be 10 times the work needed to walk up the hill).

Applying the Law of Conservation of Energy

A 10 kg rock falls from a 20 m cliff. What is the kinetic and potential energy when the rock has fallen 10 m?

Strategy

Choose the equation.

$K E_{1} + P E_{1} = K E_{2} + P E_{2}$

$\begin{matrix} K E = \frac{1}{2} m v_{}^{2}; & P E = m g h \end{matrix}$

$\frac{1}{2} m v_{1}^{2} + m g h_{1} = \frac{1}{2} m v_{2}^{2} + m g h_{2}$

List the knowns.

m = 10 kg, v₁ = 0, g = 9.80

\frac{m}{s^{2}},

h₁ = 20 m, h₂ = 10 m

Identify the unknowns.

KE₂ and PE₂

Substitute the known values into the equation and solve for the unknown variables.

Solution

$P E_{2} = m g h_{2} = 10 (9.80) 10 = 980 J$

$K E_{2} = P E_{2} - (K E_{1} + P E_{1}) = 980 - {[0 - [10 (9.80) 20]]} = 980 J$

Discussion

Alternatively, conservation of energy equation could be solved for v₂ and KE₂ could be calculated. Note that m could also be eliminated.

Practice Problems

A child slides down a playground slide. If the slide is 3 m high and the child weighs 300 N, how much potential energy does the child have at the top of the slide? (Round g to $10 m / s^{2} .$ )

0 J
100 J
300 J
900 J

A 0.2 kg apple on an apple tree has a potential energy of 10 J. It falls to the ground, converting all of its PE to kinetic energy. What is the velocity of the apple just before it hits the ground?

0 m/s
2 m/s
10 m/s
50 m/s

Converting Potential Energy to Kinetic Energy

In this activity, you will calculate the potential energy of an object and predict the object’s speed when all that potential energy has been converted to kinetic energy. You will then check your prediction.

You will be dropping objects from a height. Be sure to stay a safe distance from the edge. Don’t lean over the railing too far. Make sure that you do not drop objects into an area where people or vehicles pass by. Make sure that dropping objects will not cause damage.

You will need the following:

Materials for each pair of students:

Four marbles (or similar small, dense objects)
Stopwatch

Materials for class:

Metric measuring tape long enough to measure the chosen height
A scale

Instructions

Procedure

Work with a partner. Find and record the mass of four small, dense objects per group.
Choose a location where the objects can be safely dropped from a height of at least 15 meters. A bridge over water with a safe pedestrian walkway will work well.
Measure the distance the object will fall.
Calculate the potential energy of the object before you drop it using PE = mgh = (9.80)mh.
Predict the kinetic energy and velocity of the object when it lands using PE = KE and so, $m g h = \frac{m v^{2}}{2}; v = \sqrt{2 (9.80) h} = 4.43 \sqrt{h} .$
One partner drops the object while the other measures the time it takes to fall.
Take turns being the dropper and the timer until you have made four measurements.
Average your drop multiplied by and calculate the velocity of the object when it landed using v = at = gt = (9.80)t.
Compare your results to your prediction.

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Check Your Understanding

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True or falseIf a rock is thrown into the air, the increase in the height would increase the rocks kinetic energy, and then the increase in the velocity as it falls to the ground would increase its potential energy.

True
False

Identify equivalent terms for stored energy and energy of motion.

Stored energy is potential energy, and energy of motion is kinetic energy.
Energy of motion is potential energy, and stored energy is kinetic energy.
Stored energy is the potential as well as the kinetic energy of the system.
Energy of motion is the potential as well as the kinetic energy of the system.

Section Summary

Mechanical energy may be either kinetic (energy of motion) or potential (stored energy).
Doing work on an object or system changes its energy.
Total energy in a closed, isolated system is constant.

Key Equations

conservation of energy

K E_{1} + P E_{1} = K E_{2} + P E_{2}

Concept Items

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The starting line of a cross country foot race is at the bottom of a hill. Which form(s) of mechanical energy of the runners will change when the starting gun is fired?

Kinetic energy only
Potential energy only
Both kinetic and potential energy
Neither kinetic nor potential energy

Critical Thinking

True or falseA cyclist coasts down one hill and up another hill until she comes to a stop. The point at which the bicycle stops is lower than the point at which it started coasting because part of the original potential energy has been converted to a quantity of heat and this makes the tires of the bicycle warm.

True
False

Problems

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The potential energy stored in a compressed spring is $U = \frac{1}{2} k x^{2}$ , where k is the force constant and x is the distance the spring is compressed from the equilibrium position. Four experimental setups described below can be used to determine the force constant of a spring. Which one(s) require measurement of the fewest number of variables to determine k? Assume the acceleration due to gravity is known.

An object is propelled vertically by a compressed spring.
An object is propelled horizontally on a frictionless surface by a compressed spring.
An object is statically suspended from a spring.
An object suspended from a spring is set into oscillatory motion.

I only
III only
I and II only
III and IV only

The correct answer is (d), which features two setups where two quantities must be measured to find the spring constant. The period of an object suspended from a spring is determined by the mass of the object and the spring constant $(T = 2 π \sqrt{m / k})$ , so only the mass and period need to be measured to find the force constant, k. The distance x a spring stretches when an object is suspended from it can be determined by the equation $m g = k x$ , so only the mass and the distance need to be measured to find k. The remaining two arrangements require measurement of three quantities: the mass and a displacement plus either a height or a speed, since (I) is described by the equation $m g h = \frac{1}{2} k x^{2}$ and (II) is described by the equation $\frac{1}{2} k x^{2} = \frac{1}{2} m v^{2}$ .

Test Prep Multiple Choice

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Test Prep Short Answer

True or falseThe formula for gravitational potential energy can be used to explain why joules, J, are equivalent to kg × m² / s² . Show your work.

True
False

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True or falseA marble rolls down a slope from height h₁ and up another slope to height h₂, where (h₂ < h₁). The difference mg(h₁ h₂) is equal to the heat lost due to the friction.

True
False

Test Prep Extended Response

Acceleration due to gravity on the moon is 1.6 m/s² or about 16% of the value of g on Earth.
If an astronaut on the moon threw a moon rock to a height of 7.8 m, what would be its velocity as it struck the moons surface?
How would the fact that the moon has no atmosphere affect the velocity of the falling moon rock? Explain your answer.

The velocity of the rock as it hits the ground would be 5.0 m/s. Due to the lack of air friction, there would be complete transformation of the potential energy into the kinetic energy as the rock hits the moons surface.
The velocity of the rock as it hits the ground would be 5.0 m/s. Due to the lack of air friction, there would be incomplete transformation of the potential energy into the kinetic energy as the rock hits the moons surface.
The velocity of the rock as it hits the ground would be 12 m/s. Due to the lack of air friction, there would be complete transformation of the potential energy into the kinetic energy as the rock hits the moons surface.
The velocity of the rock as it hits the ground would be 12 m/s. Due to the lack of air friction, there would be incomplete transformation of the potential energy into the kinetic energy as the rock hits the moons surface.

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Glossary

law of conservation of energy: states that energy is neither created nor destroyed

Attribution. Physics, OpenStax, Rice University, and the contributing authors credited in the book. Licensed under Creative Commons Attribution 4.0 International. Source revision 56e71824e7 of openstax/osbooks-physics (2026-10-13). CC BY 4.0. Original source ↗. Text edition of a pinned CC BY 4.0 source revision of the OpenStax book, converted from CNXML by the EvidenceHelix importer. Original prose, learning objectives, tables, exercises, glossary and captions are preserved; images are not copied (figures link to the original source and keep their credits). No new content was added. Later revisions of this book can carry different licence terms and are not used.