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2.D Analyzing Functional Dependence

Science Practice 2 is about using mathematical routines to describe and explain phenomena, as well as to solve problems. Skill 2.D is to predict new values or factors of change of physical quantities using functional dependence between variables.

This skill will be 15-20% of the multiple-choice section of the AP Physics 1 exam and will be only part of 30-40% of the free response dedicated to all of science practice 2. The individual skills do not have a % breakdown provided for the free-response section.

How to problems that require functional dependence analysis

If the prompt does any of these things you are being asked to use functional dependence:

Asks how or whether an equation supports a claim. This is often a qualitative claim.
- example: "A student claims that increasing the height of a horizontally thrown ball will increase the distance it travels before hitting the ground. Another student determines the equation d = v*sqrt(2h/g). Explain how the equation supports the first student's claim."
Asks the value of a previously quantity based on a change. This can be a symbolic value (like F for a defined force) or a numerical value (100 N).
- example: "Energy dissipated by the frictional force as the block travels from a speed of v to stop is E. If the initial velocity of the block is doubled, the energy dissipated by the frictional force as the block comes to rest is A) 1/2E B) E C) 4E . . ."
Asks you to connect a graph showing a mathematical relationship to a physical situation with defined constants and variables.
- example: "A block is released with a speed v and comes to rest due to a constant frictional force. Which of the following graphs shows the speed as a function of time as the block comes to rest?"

If the answer choices of a MC question are in this format you are being asked to use functional dependence:

the answers are all variations of the same value, like F, 2F, 1/2F, F^2
the answers are a fraction representing a ratio between two versions of the same variable. (1/2, 1, 2)

How to approach functional dependence

First: Start with the right equation. Functional dependence is an analysis how a value changes based on its factors, so we need to have an equation solved for the responding variable that depends only on constants and values you are given change info for.

Identify the responding variable, known constants, and what values you have change information for.
If an equation is provided for you, make sure all it only has the values you identified in it. If not, or you are not given an equation, refer to Skill 2.A Derivation to develop the equation that meets the requirements underlined above.

Second, select the approach that best matches the question.

For supporting a claim qualitatively evaluate the effect of changes on the right-hand side of the equation on the overall value.

For simple fractions

- increasing the numerator increases the value of the fraction, decreasing the numerator decreases the value
- increasing the denominator decreases the value of the fraction, increasing the denominator increases the value

Evaluate subtraction and addition carefully - especially if some of the values are negative.

Example answer: 4,π,G, and M are constant in this scenario, so, T varies only based on r. Since r is in the numerator, an increase in orbital distance will increase the value of the fraction and T, the orbital period, will also increase. So, the equation does support the student's claim.

Example: A student finds this equation and says indicates that it takes more time T for an asteroid to orbit the sun increases as orbital distance r increases. In this equation M is the mass of the sun and both π and G are constants. How does this equation confirm or disprove this claim?

2. For matching to a graph match the equation to the 6 standard graph shapes at right (click through 2 slides of shapes).

- Make sure the equation is solved for the variable that is on the y axis
- Look to see if the variable on the x axis is modified in some way (in the bottom of a fraction, squared, under a square root radical sign, etc) and find the equation that has x modified in the same way.
  - It may help you to see the similarity if you rearrange the equation to group constants separate from variables.
- Check for a vertical intercept to see if it is positive, negative, or zero by substituting a zero in for the quantity on the x axis.
- Check signs to be sure if the line should go up or down.

Example: "A ball is thrown from the top of a tower at an angle of 45 degrees above the horizontal. Which of the following graphs could represent the ball's vertical velocity as a function of time?" See the graph choices and answer in the 3rd and 4th slide in the image carousel at right.

3. For an answer in ratio form write two forms of the equation.

Derive your equation for the first condition using the same symbol and subscript as the question establishes.
Rewrite your equation for the second condition using the new symbol and subscript established in the question.
Divide the left side and right side of the equations and simplify

Try and solve the example at right. If you have not learned energy yet, the energy lost is kinetic energy K, where K=1/2mv^2.

4. For an answer in terms of a previous value find the factor of change.

The factor of change method is essentially the same thing as approach 3 for ratios, except you will solve for one variable instead of a ratio. There is a shortcut to keep your math simple, though. Here is the shortcut method:

Once you have your derived equation solved for the right variable, set up parentheses under the right side of the equation and fill in the factor of change for each item in the original expression.
- Constants do not change, so we fill in 1 for them. (a common error is to fill in numerical values instead of factors when numbers are present in the equation)
- Make sure to apply the modifications of square roots, exponents, fractions to factors for changed variables
Simplify the factors in parentheses. This is the factor of change for the left side of the equation too.
Write your new equation in this format: [symbol for new value] = [factor of change found in parentheses][symbol for old value]

Try the example at right and check the answer on slide 2.

Practice

Proportions in Kinematics Practice: Use factor of change to find an answer in terms of a previous value
Find the QQT (Quantitative Qualitative Translation) questions from past exams. From 2015-2024 these were either question 2 or 3.
Practice with sketching graphs on every problem you attempt. If no change has been proposed, simply ask, "what relationship exists between the answer and this particular variable (select one from the equation)?" and then sketch the graph, considering the intercept/starting condition, + or - trend, and the nature od the curve or straight line.

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