6 min readβ’november 2, 2020

Peter Apps

The **Qualitative / Quantitative Translation **is one of the 5 FRQs that appear on the AP Physics 1 Exam. This particular question is usually worth 12 points (out of a total of 45) and is suggested to take around 25 minutes to complete. The word **qualitative implies the use of words**. The word **quantitative implies the use of mathematics**. This means that a QQT is a question that requires you to go between words and mathematics in describing and analyzing a situation.

- Does it have a slope (positive/negative/zero) that makes sense?
- Ex. An object that is moving in the positive direction and speeding up should have a positive slope

- Does it have an intercept (on the y-axis or x-axis or none at all) that makes sense?
- Ex. A Period vs Length graph for a pendulum should have a 0,0 intercept.
- Ex. A velocity vs time graph should have a y-intercept equal to the initial velocity

- Should the graph be a line or a curve?
- Is the relationship linear (Fs = kx) or quadratic (Us = 1/2kx^2)?

- The symbols in the numerator - when they increase, does the quantity of interest increase?
- The symbols in the denominator - when they increase, does the quantity of interest decrease?
- Should a variable be inversely (one decreases if another increases, y=k/x) or directly related (one increases if another increases, y=1/2ax^2)?

- If the student uses proportional reasoning (sounds like this: βif the speed of a car doubles, then it has twice the energyβ) check the equation that shows this relationship (in this case, K = 1/2mv^2) and see if there is actually a direct relationship (in this case, no it's a quadratic relationship). You would need to correct the student on this (it's actually four times the energy).
- If the student says that something increases or decreases (sounds like this: βif the ball moves faster, then the time it takes to pass through the photogate increasesβ), check to see if that makes sense (βno it doesnβtβa faster ball would spend less time passing through a photogate). Get ready to correct the studentβs wrong reasoning.

The student should aim to the right of point C. In order to change the angular speed of the rod, a torque needs to be applied to it. The amount of torque depends on the force and the distance from the pivot. Increasing the distance from the pivot increases the torque which will increase the angular speed.

This equation agrees with my reasoning in part (a). If the distance from the pivot (*x)* is increased the angular speed π increases as well.

This equation shows that increasing the mass of the disk would result in a slower angular speed because mdisk is in the denominator. But a more massive disk has more angular momentum to transfer to the rod and should result in a larger angular speed. Also, a larger moment of inertia *I *makes the rod more difficult to rotate and should give a smaller angular speed not a larger one, so *I should be in the denominator not the numerator*.

The disk bouncing off the rod has a larger change in angular momentum than when it sticks to the rod. This results in a larger angular speed in the rod afterwards because the total angular momentum of the rod-disk system has to remain constant.

(From National Math + Science Initiative)

Two cases are shown:

Case 1 in which the block on the table is initially moving to the right, and Case 2 in which the block on the table is initially moving to the left. The two students are arguing over which case has greater acceleration and which case has greater tension in the string.

i. Which aspects of Student Xβs reasoning are correct? Explain why they are correct.

Student X correctly states that both cases have the same hanging mass (shown in diagram), same force of friction on the table (Same mass on the table and same coefficient of friction, so Ff = πFn will be the same)
ii. Which aspects of Student Xβs reasoning are incorrect? Explain why they are wrong.

The forces between the 2 trials are not the same. In case 1 the frictional force and the weight of the hanging block are in the same direction (right / down) and result in a larger net force. This leads to a larger acceleration for case 1. The tension on the string in case 1 is smaller, since the acceleration is greater for the hanging block (mg-T=ma) In case 2, the frictional force is to the left and is opposing the weight of the hanging block, which gives a smaller net force and acceleration. The tension is greater in case 2 as well.
iii. Which aspects of Student Yβs reasoning are correct? Explain why they are correct.

Student Y correctly states that the frictional force changes direction (opposes the motion of the blocks) which gives a greater net force & acceleration in Case 1 (see reasoning in part ii for more detailed explanation)
iv. Which aspects of Student Yβs reasoning are incorrect? Explain why they are wrong

Greater acceleration doesn't mean greater tension in the cord. In fact the opposite is true. Looking just at the hanging block, we can derive a Newton's 2nd Law equation to show this.

F = ma

mg - T = ma

T = mg - ma <-- a larger acceleration means a smaller tension!

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