Problem Solving With the Conservation of Energy

To solve a conservation of energy problem, determine the system of interest, apply law of conservation of energy, and solve for the unknown.

Key Points

• If you know the potential energies for the forces that enter into the problem, then forces are all conservative, and you can apply conservation of mechanical energy simply in terms of potential and kinetic energy. The equation expressing conservation of energy is: KEi+PEi=KEf+PEf.

• If you know the potential energy for only some of the forces, then the conservation of energy law in its most general form must be used: KEi+PEi+Wnc+OEi=KEf+PEf+OEf, where OE stands for all other energies.

• Once you have solved a problem, always check  the answer to see if it is reasonable.

Terms

• The energy an object has because of its position (in a gravitational or electric field) or its condition (as a stretched or compressed spring, as a chemical reactant, or by having rest mass)

• The energy possessed by an object because of its motion, equal to one half the mass of the body times the square of its velocity.

• A force with the property that the work done in moving a particle between two points is independent of the path taken.

Figures

1. Determining Energy

The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. When they start rising, the kinetic energy begins to be converted to gravitational potential energy. The sum of kinetic and potential energy in the system remains constant, ignoring losses to friction.

2. Energy conservation

Part of a series of videos on physics problem-solving. The problems are taken from "The Joy of Physics." This one deals with energy conservation. The viewer is urged to pause the video at the problem statement and work the problem before watching the rest of the video.

Problem-solving Strategy

You should follow a series of steps whenever you are problem solving.

Step One

Determine the system of interest and identify what information is given and what quantity is to be calculated. For example, let's assume you have the problem with car on a roller coaster. You know that the cars of a roller coaster reach their maximum kinetic energy ($KE$) when at the bottom of their path. When they start rising, the kinetic energy begins to be converted to gravitational potential energy ($PE_g$). The sum of kinetic and potential energy in the system should remain constant, if losses to friction are ignored (Figure 1).

Step Two

Examine all the forces involved and determine whether you know or are given the potential energy from the work done by the forces. Then use step three or step four.

Step Three

If you know the potential energies ($PE$) for the forces that enter into the problem, then forces are all conservative, and you can apply conservation of mechanical energy simply in terms of potential and kinetic energy. The equation expressing conservation of energy is: $KE_i+PE_i=KE_f+PE_f$.

Step Four

If you know the potential energy for only some of the forces, then the conservation of energy law in its most general form must be used:

$KE_i+PE_i+W_{nc}+OE_i=KE_f+PE_f+OE_f$

where $OE$ stand for all other energies. In most problems, one or more of the terms is zero, simplifying its solution. Do not calculate $W_c$, the work done by conservative forces; it is already incorporated in the $PE$ terms.

Step Five

You have already identified the types of work and energy involved (in step two). Before solving for the unknown, eliminate terms wherever possible to simplify the algebra. For example, choose height $h = 0$ at either the initial or final point—this will allow to set $PE_g$ at zero. Then solve for the unknown in the customary manner.

Step Six

Check the answer to see if it is reasonable. Once you have solved a problem, reexamine the forms of work and energy to see if you have set up the conservation of energy equation correctly. For example, work done against friction should be negative, potential energy at the bottom of a hill should be less than that at the top, and so on.

Figure 2

Key Term Glossary

conservation
A particular measurable property of an isolated physical system does not change as the system evolves.
Appears in these related concepts:
conservative force
A force with the property that the work done in moving a particle between two points is independent of the path taken.
Appears in these related concepts:
energy
A quantity that denotes the ability to do work and is measured in a unit dimensioned in mass × distance²/time² (ML²/T²) or the equivalent.
Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.
Appears in these related concepts:
force
A physical quantity that denotes ability to push, pull, twist or accelerate a body which is measured in a unit dimensioned in mass × distance/time² (ML/T²): SI: newton (N); CGS: dyne (dyn)
Appears in these related concepts:
Force
A force is any influence that causes an object to undergo a certain change, either concerning its movement, direction or geometrical construction.
Appears in these related concepts:
friction
A force that resists the relative motion or tendency to such motion of two bodies in contact.
Appears in these related concepts:
kinetic
Of or relating to motion
Appears in these related concepts:
kinetic energy
The energy possessed by an object because of its motion, equal to one half the mass of the body times the square of its velocity.
Appears in these related concepts:
Kinetic Energy
The energy associated with a moving particle or object having a certain mass.
Appears in these related concepts:
Law
A concise description, usually in the form of a mathematical equation, used to describe a pattern in nature
Appears in these related concepts:
potential
A curve describing the situation where the difference in the potential energies of an object in two different positions depends only on those positions.
Appears in these related concepts:
potential energy
The energy an object has because of its position (in a gravitational or electric field) or its condition (as a stretched or compressed spring, as a chemical reactant, or by having rest mass)
Appears in these related concepts:
quantity
A fundamental, generic term used when referring to the measurement (count, amount) of a scalar, vector, number of items or to some other way of denominating the value of a collection or group of items.
Appears in these related concepts:
series
A number of things that follow on one after the other or are connected one after the other.
Appears in these related concepts:
work
A measure of energy expended in moving an object; most commonly, force times displacement. No work is done if the object does not move.