AP Physics C Mechanics 2.9 Resistive Forces Overview

[Maximum number: 15]

A student drops a sphere of mass m from rest. The air exerts a drag force of magnitude $F_{\text {drag }}$ on the sphere, as shown in Figure 1. The student models the magnitude of the drag force as $F_{\text {drag }}=b v$ , where v is the speed of the sphere and b is a positive constant with appropriate units.

(a)

Derive, but do NOT solve, a differential equation that could be used to determine the speed v of the sphere as a function of time t. Express your answer in terms of given quantities and physical constants, as appropriate.

Figure 2

[ 3 ]

For a multi-step derivation that includes Newton's second law of motion

1 point

For indicating that the net force exerted on the sphere includes only the gravitational force and
a drag force

1 point

Example Response
$F_{\mathrm{net}}=F_{g}-F_{\mathrm{drag}}$

For a correct differential equation that is in terms of the given variables

Scoring Note: Variables do not have to be separated for this point to be earned.

1 point

Example Response
$m \frac{d v}{d t}=m g-b v$

Example Solution
$\begin{aligned} \Sigma F=m a F_{g}-F_{\mathrm{drag}}=m a m g-b v=m a m \frac{d v}{d t}=m g-b v \end{aligned}$

Total for part (a)

3 points

(b)

The student sketches the drag force $F_{\text {drag }}$ exerted on the sphere as a function of time t, as shown in Figure 2.

[ 3 ]

(i)

For a vertical line labeled $t_{\mathrm{T}}$ at the approximate location at which the line becomes horizontal
Example Response

1 point

(i)

Draw a vertical line on the sketch in Figure 2 to indicate the earliest time at which $F_{\text {drag }}$ is equal to the magnitude of the weight of the sphere, which occurs when the sphere reaches terminal speed. Label this time as $t_{\mathrm{T}}$ on the time axis.

[ 1 ]

For a vertical line labeled $t_{\mathrm{T}}$ at the approximate location at which the line becomes horizontal
Example Response

1 point

(ii)

Justify the location of $t_{\mathrm{T}}$ . Explicitly reference appropriate features of the sketch in Figure 2.

[ 2 ]

For a response that references the slope of the graph or the rate at which the slope changes

1 point

For correctly relating a feature of the graph to the forces exerted on the sphere as the sphere reaches terminal speed

1 point

Example Response

For the times leading up to $t_{\mathrm{T}}$ , the slope of the graph is positive which means that the

magnitude of the drag force is still increasing. After $t_{\mathrm{T}}$ , the slope of the graph is zero which

means that the magnitude of the drag force is constant and equal to the downward

gravitational force, which indicates that the net force is zero and that the sphere has reached

a constant terminal velocity.

Total for part (b)

3 points

(c)

Suppose the student throws the same sphere downward with a nonzero initial speed. The magnitude of the new drag force at terminal speed after being thrown downward is $F_{\text {new }}$ .

Indicate whether $F_{\text {new }}$ would be greater than, less than, or equal to the magnitude of $F_{\text {drag }}$ at terminal speed represented in Figure 2. Greater than Less than Equal to

Briefly justify your answer.

[ 2 ]

(d)

The student conducts an experiment to better understand the relationship between $F_{\text {drag }}$ and v. The student makes measurements to calculate and graph the magnitude of $F_{\text {drag }}$ as a function of v for the falling sphere.

$F_{\text {drag }}(\mathrm{N})$

[ 4 ]

(i)

Draw the best-fit line for the data.

[ 1 ]

(ii)

Use the best-fit line to calculate an experimental value for b.

A student claims that the terminal speed $v_{\mathrm{T}}$ of the sphere depends on the diameter D of the sphere. The student designs an experiment to collect data that can be used to provide evidence to support the claim.

[ 3 ]

(ii) For calculating a value for the slope of the line using two points on the best-fit line 1 point
Scoring Note: Using data points that fall on the best-fit line earns this point.
Example Response

\text { slope }=\frac{15 \mathrm{~N}-5 \mathrm{~N}}{21 \mathrm{~m} / \mathrm{s}-5 \mathrm{~m} / \mathrm{s}}

For using the correct relationship between the slope of the best-fit line and the value of b 1 point
Example Response

\begin{aligned} & \text { slope }=\frac{\left|F_{\text {drag }}\right|}{v} \\ & \text { slope }=b \end{aligned}

For a calculated value of b that is $0.6 \mathrm{~kg} / \mathrm{s}<b<0.8 \mathrm{~kg} / \mathrm{s}$ 1 point
Example Response

b=0.625 \mathrm{~kg} / \mathrm{s}

Example Solution

\begin{aligned} & \left|F_{\text {drag }}\right|=b v \\ & \frac{\left|F_{\text {drag }}\right|}{v}=b \\ & \text { slope }=b \\ & b=\frac{15 \mathrm{~N}-5 \mathrm{~N}}{21 \mathrm{~m} / \mathrm{s}-5 \mathrm{~m} / \mathrm{s}} \\ & b=0.625 \mathrm{~kg} / \mathrm{s} \end{aligned}

(e)

The student has access to but does not have to use all of the following equipment.

- Sphere Set 1: spheres of the same known mass with different known diameters

- Sphere Set 2: spheres of the same known diameter with different known masses

- A motion detector that can measure velocity as a function of time

[ 3 ]

(i)

Indicate two quantities that when graphed could be used to determine whether the diameter of the sphere affects the terminal speed.

Vertical axis: Horizontal axis:

[ 2 ]

(ii)

ii. Briefly describe how the quantities graphed could be used to determine the relationship between sphere diameter and terminal speed.

Figure 1 Note: Figure not drawn to scale.

[ 1 ]

Covered syllabus points

2.9 Resistive Forces question 1

Question (a)

Question (b)

Question (i)

Question (ii)

Question (c)

Question (d)

Question (i)

Question (ii)

Question (e)

Question (i)

Question (ii)