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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 2.11 Sensemaking: Covariational Reasoning

You previously learned how to make sense of both units and numbers in Sensemaking: Units and Numbers. Perhaps the most powerful type of physics sensemaking is symbolic, since so much of physics is concerned with symbolic equations. The most general way to make sense of a symbolic equation is with covariational reasoning.
When you use covariational reasoning, continue to follow the Sensemaking Steps.

Example 2.11.2. Covariational Sensemaking: Area of a Circle.

Consider the equation for the area of a circle in terms of its radius: \(A_{circle} = \pi r^2\text{.}\)
Physical Understanding: You might start by considering all the various circles you have drawn or seen. You might even draw a few circles with different radii to see how they behave. Together, physical reasoning tells you that increasing the radius of a circle will also lead to an increase in the area, since radius and area are both measures of the size of a circle.
Symbolic Equation: \(A_{circle}\) clearly depends on \(r\text{;}\) in fact, increasing \(r\) leads to an increase in \(A_{circle}\text{.}\) You can actually say something stronger: the equation is quadratic, so doubling \(r\) actually increases \(A_{circle}\) by a factor of four!
Do they agree? In this instance, the physical reasoning and the equation clearly agree with each other, as both suggest an increase in \(A_{circle}\text{.}\) Even the stronger claim agrees, because changing the radius changes the size of the circle both horizontally and vertically, so it should impact the area more strongly than a linear relationship!

Example 2.11.3. Covariational Sensemaking: Displacement.

Consider the equation \(\Delta x = v\Delta t\text{,}\) rewritten from the definition of average velocity. This equation relates \(\Delta x\) to two other quantities: velocity \(v\) and time \(\Delta t\text{.}\) We can investigate each relationship with \(\Delta x\) separately by holding the other quantity constant.
Physically, velocity is a measure of how fast something is traveling. In general, a faster object with travel a greater distance in the same amount of time than a slower object: you might think about a race or a car chase. On the other hand, two objects with the same speed can travel different distances if they travel for different times, and the more time spent traveling, the farther something will travel. The equation agrees with both because \(\Delta x\) is the product of \(v\) and \(\Delta t\text{.}\) If either \(v\) or \(\Delta t\) had been divided instead, the relationship would be inverted instead.

Exercises Activities: Practice Sensemaking

1. Rejecting Incorrect Claims.

The statements below are incorrect. For each statement, use a sensemaking strategy to determine why it is incorrect, such as the one introduced in this section or those in Sensemaking: Units and Numbers.
  1. Your friend has built a rocket and reports that it has a top speed of \(v=700 \mathrm{~m/s^2}\text{.}\)
  2. The Wizard of Oz claims the volume of his balloon \(V\) is related to the temperature \(T\) by a constant \(\alpha\text{:}\) \(V = \frac{\alpha}{T}\text{.}\)
  3. A car dealer claims that a certain new car has a mass of \(4000 \mathrm{~g}\text{.}\)
  4. A manufacturer claims that an office space with length \(x\) and width \(y\) can fit the following number of their cubicles: \(N=5\frac{y}{x}\text{.}\)

2. Metacognition.

When you are making sense of something in physics, it can be tempting to just say “Yes, this makes sense to me.”
  • Why do you think you should say more when sensemaking?
  • Why do you think covariational reasoning is useful?
  • Why do you think sensemaking in general might be useful?

3. The Ball and the Bounce I.

At \(t_o = 0 \mathrm{~s}\text{,}\) The instructor throws a tennis ball directly downward at \(v_i = 12 \mathrm{~m/s}\text{.}\) At \(t_1 = 0.2 \mathrm{~s}\text{,}\) after falling \(h = 1.2 \mathrm{~m}\text{,}\) the ball hits the ground and bounces. At \(t_2 = 0.48 \mathrm{~s}\text{,}\) the instructor catches the ball at the same point where they released it, moving upward at \(v_f = 4 \mathrm{~m/s}\text{.}\) Determine the average acceleration of the tennis ball. Do you think it would be appropriate or necessary to use a calculator for this activity?
Answer.
\begin{equation*} a_{ave} = \frac{v_f + v_i}{t_2} \end{equation*}

4. The Ball and the Bounce II.

Use covariational reasoning for each quantity to evaluate the symbolic equation for \(a_{ave}\text{.}\)
  • \(\displaystyle v_i\)
  • \(\displaystyle h\)
  • \(\displaystyle t_1\)
  • \(\displaystyle t_2\)
  • \(\displaystyle v_f\)
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