Problem Solving With the Conservation of Energy

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 .

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.

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, and $W_{nc}$ stands for work done by non-conservative forces. 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.