Problem-Solving Basics for One-Dimensional Kinematics

Read this text for a step-by-step guide on how to solve problems of constant acceleration using kinematic equations.

Unreasonable Results

Physics must describe nature accurately. Some problems have results that are unreasonable because one premise is unreasonable or because certain premises are inconsistent with one another. The physical principle applied correctly then produces an unreasonable result. For example, if a person starting a foot race accelerates at 0.40 \mathrm{~m} / \mathrm{s}^{2} for 100 s, his final speed will be 40 m/s (about 150 km/h) - clearly unreasonable because the time of 100 s is an unreasonable premise.

The physics is correct in a sense, but there is more to describing nature than just manipulating equations correctly. Checking the result of a problem to see if it is reasonable does more than help uncover errors in problem solving - it also builds intuition in judging whether nature is being accurately described.

Use the following strategies to determine whether an answer is reasonable and, if it is not, to determine what is the cause.

Step 1

Solve the problem using strategies as outlined and in the format followed in the worked examples in the text. In the example given in the preceding paragraph, you would identify the givens as the acceleration and time and use the equation below to find the unknown final velocity. That is,

v=v_{0}+a t=0+\left(0.40 \mathrm{~m} / \mathrm{s}^{2}\right)(100 \mathrm{~s})=40 \mathrm{~m} / \mathrm{s}.


Step 2

Check to see if the answer is reasonable. Is it too large or too small, or does it have the wrong sign, improper units, …? In this case, you may need to convert meters per second into a more familiar unit, such as miles per hour.

\left(\frac{40 \mathrm{~m}}{\mathrm{~s}}\right)\left(\frac{3.28 \mathrm{ft}}{\mathrm{m}}\right)\left(\frac{1 \mathrm{mi}}{5280 \mathrm{ft}}\right)\left(\frac{60 \mathrm{~s}}{\min }\right)\left(\frac{60 \mathrm{~min}}{1 \mathrm{~h}}\right)=89 \, \mathrm{mph}

This velocity is about four times greater than a person can run - so it is too large.


Step 3

If the answer is unreasonable, look for what specifically could cause the identified difficulty. In the example of the runner, there are only two assumptions that are suspect. The acceleration could be too great or the time too long.

First look at the acceleration and think about what the number means. If someone accelerates at 0.40 \mathrm{~m} / \mathrm{s}^{2}, their velocity is increasing by 0.4 \mathrm{~m} / \mathrm{s} each second. Does this seem reasonable? If so, the time must be too long. It is not possible for someone to accelerate at a constant rate of 0.40 \mathrm{~m} / \mathrm{s}^{2} for 100 \mathrm{~s} (almost two minutes).