AP Physics C Mechanics Resistive Forces FRQ

QUESTION 

A student drops a cylinder of mass m from rest. The air exerts a drag force of magnitude $F_{drag}$ on the cylinder, as
 drag shown in Figure 1. The student models the magnitude of the drag force as $F_{drag}=bv^{2}$ , where v is the speed of the
cylinder, 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 cylinder
as a function of time t. Express your answer in terms of given quantities and physical constants, as appropriate.

(b) The student correctly sketches the speed v of the cylinder as a function of time t, as shown in Figure 2.
i. Draw a vertical line on the sketch in Figure 2 to indicate the earliest time at which $F_{drag}$ on the cylinder
is equal to the magnitude of the weight of the cylinder. Label this time as$t_{1}$ on the time axis.

ii. Justify the location of $t_{1}$. Explicitly reference appropriate features of the sketch in Figure 2.

(c) Rather than dropping the cylinder from rest, the student throws the cylinder upward with a nonzero initial
speed. The cylinder is in the same orientation as when the cylinder was previously dropped. The student allows
the cylinder to fall toward the ground.
Indicate whether the magnitude of the cylinder’s maximum downward speed after being thrown upward would be
greater than, less than, or equal to the maximum speed $v_{max}$ in Figure 2.
_____ Greater than _____ Less than _____ Equal to
Briefly justify your answer.

(d) The student conducts an experiment to better understand the relationship between maximum speed $v_{max}$ and
mass. The student collects data to determine the maximum speed for cylinders dropped from rest, each with
the same physical size and shape but a different mass m. The student then graphs $v_{max}^{2}$ as a function of mass.

i. Draw the best-fit line for the data.
ii. Use the best-fit line to calculate an experimental value for B.

A student claims that the magnitude of the maximum speed of a cylinder dropped from rest depends on the length
of the cylinder. The student designs an experiment to collect data that can be used to provide evidence to support
the claim. The student drops cylinders with the orientation shown in Figure 3.
(e) The student has access to but does not have to use all of the following equipment.

• Cylinder Set 1: cylinders of the same known length with different known masses
• Cylinder Set 2: cylinders of the same known mass with different known lengths
• A motion detector that can measure velocity as a function of time

i. Indicate two quantities that when graphed could be used to determine whether the length of the cylinder affects the maximum speed.
Vertical axis: ______________ Horizontal axis: _______________

ii. Briefly describe how the quantities graphed could be used to determine the relationship between
cylinder length and maximum speed.

▶️Answer/Explanation

Ans:-

(a) For a multi-step derivation that includes Newton’s second law of motion 

Example Solution

$\sum F=ma$
$F_{g} -F_{drag}=ma_{y}$
$mg-bv^{2}=ma_{y}$
$m\frac{dv}{dt} =mg-bv^{2}$

(b)(i) For a vertical line labeled t¹ at the approximate location at which the line becomes horizontal

(b)(ii) For relating $t_{1}$ to the time at which the velocity versus time graph is constant or the slope of the line is zero 
Example Response
Because the sketched line is horizontal after t¹, the velocity is constant. If the velocity is
constant, then the acceleration is zero. Therefore, the net force is zero, which means that the
gravitational and drag forces are equal in magnitude.

(c) For selecting “Equal to” with an attempt at a relevant justification  
Example Response
At its peak, the cylinder will have a speed of 0 m/s. Therefore, the cylinder would reach the
same vmax as if the student had dropped the cylinder from rest at that height. Because the
cylinder reached vmax from the initial drop, the two max speeds are equal.

(d)(i) For drawing an appropriate line of best fit that approximates the data

(d)(ii) For calculating a value for the slope of the line using two points on the best-fit line

Scoring Note: Using data points that fall on the best-fit line is acceptable.

Example Solution

(e)(i) For indicating that the length of the cylinder should be graphed  
(e)(ii) For describing how the quantities graphed are related to the conclusions of the experiment 
Example Response
The slope of the length vs. maximum velocity graph can be used to determine if length affects terminal velocity.