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Guide 1-2a. Solving Physics Problems

We expect you to follow a tried and true method for solving physics problems. The method focuses the problem solver's attention on the facts and goal of the problem. The general steps of the process are the following.

 Read Always begin by reading the problem carefully.  Make sure you understand what it's asking for. Represent Decide from the problem statement what is given and what the goal is.  Assign algebraic symbols to the variables.  Draw a diagram to represent the situation. Plan Formulate a strategy for solving the problem. Solve Carry out your strategy in a logical fashion. The end result is typically an equation for the unknown in algebraic form. Apply If numerical information is given, substitute that information for the known variables and obtain a numerical result. Check Determine, to the best of your ability, that your answer makes sense.

As an example of this method, consider the following problem from your text. The different steps in the process of solving the problem will be presented.

The Problem: You drive 4.00 mi at 30.0 mi/h and then another 4.00 mi at 50.0 mi/h. What is your average velocity for the trip? (Note that the use of additional zeros beyond the decimal point in the numbers is to alert you to the fact that you're to use 3 significant figures in calculations.)

Represent

List all the Given information.

Identify each item of numerical information and assign it a symbol. Whenever possible, use the same symbols that are used in the text (x for position, Δx for displacement, v for velocity, t for time). Use subscripts to distinguish similar quantities.

Δx1 = 4.00 mi
Δx2 = 4.00 mi
v1 = 30.0 mi/h
v2 = 50.0 mi/h

List any other information or assumptions that you think you'll need. In this case, you need to assume the following:

Ignore the amount of time it takes the car to speed up from 30.0 to 50.0 mi/h.

State the Goal.

What is the average velocity, vav, for the trip?

Draw a Diagram displaying the relevant information. For 1-dimensional problems such as this one, show the origin and direction of the x-axis that you select. You don't have to actually show the car, but you may do so if you wish. Be sure, though, to make your sketches LARGE so that the details can be seen easily.

Note that three positions of the car are marked at 0, +4.00, and +8.00 miles. The positions are all measured from the Start, which is taken to be the origin. The positions are positive numbers, because the direction of +x is taken to be to the right. Setting up an axis like this is very important in physics problems, and is a key to avoiding sign mistakes in problems.

At this point, go back and make sure that the signs are correct on the given information.  Δx1 and Δx2 represent displacements. Each is the difference of a final and an initial position. Since all positions are positive and the car is traveling right, the displacements are positive.  (xf > xi) This means the velocities will also be positive. Recall that average velocity is Δxt. We've seen that Δx is positive, and Δt is always positive. (That is, time always increases.)

Plan

We know that average velocity is displacement divided by elapsed time. We'll have to add the displacements for the two parts of the trip and divide that result by the sum of the time intervals. We'll determine the time interval for each part of the trip by applying the average velocity formula to each part separately.

Solve

 The competent problem solver solves a problem algebraically before substituting numbers. Begin by writing the formulas symbolically. Then solve algebraically for the unknown before substituting any numbers. Numbers are only substituted at the last step. There are several reasons for this. One of them is that by coming up with a symbolic equation, you have actually solved all problems of the same type. You just substitute in different numbers to get a new results. Another reason is that the symbolic equation provides a way to check your answer through dimensional analysis and other means. A third reason is that you minimize rounding error. When you substitute numbers in intermediate steps, each step introduces rounding error. The error propagates with each step.

In order to calculate average velocity, you need this formula:

We'll take the initial time to be 0; however, we don't know the final time. So we need a formula to find that.  We can use the same formula after rearranging it.

We need to apply the formula to each half of the trip. For the first half,

The total elapsed time of the trip is:

Apply

Continuing with the problem above, since we're interested in the entire trip, we'll take xi to be the Start at 0 and xf to be the End at 8.00 mi. Now we're ready to substitute the given information. Always include the units.

Note that we didn't write intermediate results of calculation. We just did the calculation all at once in a calculator. This minimizes round-off error.

Check

We're not done yet. A good problem solver always checks his or her work. The following three checks should always be done.

Units check:  Make sure that the units reduce correctly.  Look at the final formula for vav. On the left, you have mi/h for velocity.  In the denominator on the right, you have mi ÷ (mi/h), which reduces to h. So the units on the right are also mi/h. Thus the units check works.

Sign check:  The final result for average velocity is positive. That makes sense because positive was defined to the right.

Sensibility check:  Does the result make sense? The answer is closer to 30 mi/h than 50 mi/h. That makes sense, because the car spends more time at the lower speed.

The solution is done.  Would we expect all of the above to appear on your paper? Click here to see a solution in the form that we would expect it from a student.