# University of Wisconsin Green Bay

### Conservation of Energy

Conservation of Mechanical Energy problems relate speed of an object at different positions. In order to work a problem using Conservation of Energy, you need to know either that there are no significant forces taking energy out of the system or the size of those forces. Conservation of Energy will not tell you about the time it takes to go between two positions. Conservation of Energy can also be used to track thermal energy for systems that change temperature.

#### How to Solve Energy Problems

• ##### 1.Identify the Problem

Energy is never created or destroyed, although under action of a force it can change in form. Therefore, the idea of conservation of energy always applies. It is only useful for solving a problem, however, when you have enough information to track the changes in energy. You can track changes in mechanical energy (problems involving motion) if either there are no significant non-conservative forces present or if you are given adequate information about those forces.

You can also use the conservation of energy in thermal energy problems if you are asked to relate changes in thermal energy to temperature changes within, or work done by, a system.

• ##### 2. Draw a Picture

Mechanical energy problems ask you to relate the speed of your system at different positions. Therefore, the most useful picture is a sketch of the actual motion, with all known speed and position information labeled.

For thermal energy problems, changes take place in properties that can’t be drawn well (such as temperature) and so a picture is not always relevant to organize the information.

• ##### 3. Select the Relation

There are two ways to begin a mechanical energy problem. The first is to begin with the equation

KE1 + PE1 = KE2 + PE2 - Wnc

and to fill into all relevant terms. You can also begin with an energy chain (track the energy throughout the problem) and write a term in your equation for each term in the chain. You will arrive at the same results.

For thermal energy problems, you will often begin with Conservation of Energy stated as

ΔQ = ΔU + W

although, again, an energy chain may be useful (especially for problems in which you look at thermal energy going from one part of the system to another.)

• ##### 4. Solve the Problem

Once you have drawn your picture and selected your relation, solving a Conservation of Energy problem is merely a matter of doing algebra. Energy is not a vector, although signs do carry meaning and so cannot be ignored.

##### 5. Understand the Results

The best way to understand what is happening in an energy problem is to draw an energy chain. Start with the form(s) of energy at the initial point in the problem, and track that energy at each subsequent point of interest. Make sure to use arrows to the side to show energy leaving the system. You can then relate that chain both to a narrative of the problem and to the equation that you have after all zeros have been filled in. As always, make sure that your answer makes physical sense.

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#### How to Solve Momentum Problems

• ##### 1.Identify the Problem

As a quick rule of thumb, if your problem involves the collision of two objects, or the separation of one system into parts, then the momentum of the combined system of all objects or parts is conserved over the time of the collision or separation.

The physics behind this rule of thumb comes directly from Newton’s Second Law. From the 2nd Law, you can see that change in momentum (Δp) is equal to FnetΔt. So if the net force on a system is zero, or small enough that FnetΔt ≈ 0 for the time period of interest, then Δp ≈ 0 and momentum is conserved. A collision or separation tends to take place over a very short time interval, so for smaller forces like friction and gravity FΔt ≈ 0 is usually true. The force involved in the collision or separation itself, however, is large. That force cancels out (Newton’s Third Law) for the system as a whole, but of course each piece feels the force and has a usually significant change in its momentum.

• ##### 2. Draw a Picture

In Conservation of Momentum problems, you compare momentum (mv) of a system before and after a collision or separation. Therefore, you want to draw a picture of the system just before the interaction and another picture just after the interaction. Label m and v information for both. Because velocity is a vector, make sure to indicate the direction of motion as well as the speed.

• ##### 3. Select the Relation

All Conservation of Momentum problems are understood by;

psystem before the interaction = psystem after the interaction

Momentum terms need to be included for each piece of the system and signs are given by the direction of velocity.

• ##### 4. Solve the Problem

In many cases, collisions are one dimensional and so solving the problem is just a matter of algebra as long as you have carefully put in the signs of velocity. For two dimensional problems, you need to divide momenta into their x- and y-components and solve each equation separately.

• ##### 5. Understand the Results

You can best understand momentum problems by tracking the momentum. Which objects sped up? Where did that momentum come from? Which slowed down? Where did it go?

#### Help! I can’t find an example that looks like the problem I need to work!

• ##### Are you certain your problem is an energy or momentum problem?

One of the most common mistakes is to think too hard. If you are told velocity and asked for kinetic energy, for example, you aren’t tracking energy changes and don’t need to go through all of the steps of a Conservation Law problem. Check Definition and Ratio problems to see if you can find a useful example.

It is also possible that your problem is better solved using kinematics (description of motion). This is rare as energy is almost always an easier approach than kinematics. However, if you are asked for time you may have a kinematics problem.

• ##### Yes, my problem is definitely an energy or momentum problem.

And every single energy problem in this section uses the very same approach, as does every momentum problem. This means any problem within those categories is an appropriate example to help you approach your problem. It isn’t the way a problem looks that determines how you solve it, it is the type of interaction (in this case, energy or momentum) that you need to consider.

That said, different situations require you to do different side problems along the way. Many problems involve using both energy and momentum, and there are a variety of question types in thermal energy problems. So if your problem has any of these features, you may find it useful to pick an example that does as well. But don’t worry, you don’t need (or want) an example to look exactly like your problem!