AP Physics C Mechanics - 2.9 Resistive Forces- Exam Style questions- FRQs

Resistive Forces AP  Physics C Mechanics FRQ

Unit 2: Force and Translational Dynamics

Weightage : 20-15%

AP Physics C Mechanics Exam Style Questions – All Topics

Question

A student drops a cylinder of mass \(m\) from rest. The air exerts a drag force of magnitude \(F_{\text{drag}}\) on the cylinder, as shown in Figure \(1\). The student models the magnitude of the drag force as \(F_{\text{drag}}=bv^{2}\), 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 \(F_{\text{drag}}\) 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 \(v_{\text{max}}\) 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 \(v_{\text{max}}\) 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 \(v_{\text{max}}^{2}\) 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.

Most-appropriate topic codes (AP Physics C: Mechanics):

• Topic \(2.9\) — Resistive Forces (Parts \( \mathrm{a} \), \( \mathrm{b} \), \( \mathrm{c} \), \( \mathrm{d} \))
• Topic \(2.5\) — Newton’s Second Law (Part \( \mathrm{a} \))
▶️ Answer/Explanation

(a)
To find the differential equation, we can start by writing out Newton’s second law for the cylinder falling downward. It’s acted upon by gravity pulling it down and a drag force resisting its motion upwards.
\( \Sigma F_y = ma_y \)
\( F_g – F_{\text{drag}} = ma_y \)
\( mg – bv^2 = m\dfrac{dv}{dt} \)

(b)(i)


On Figure 2, draw a vertical dashed line down to the horizontal time axis originating exactly at the point where the curve turns completely horizontal. Label this specific time as \(t_1\).

(b)(ii)
Looking at the graph, the line becomes totally horizontal after \(t_1\), which means the velocity has stopped changing and is now constant. A constant velocity tells us that the cylinder’s acceleration has dropped to zero. For the acceleration to be zero, the net force must also be zero—meaning the upward drag force has grown large enough to perfectly balance the downward pull of gravity.

(c)
Correct answer: Equal to

If the student throws the cylinder straight up, it will eventually slow down, momentarily stop at its peak, and then begin falling. At that highest point, its speed is \(0\,\text{m/s}\), which is exactly the same starting condition as when it was dropped from rest. Since it reached \(v_{\text{max}}\) from the lower drop height, falling from an even higher peak guarantees it will have plenty of time to hit that same terminal velocity again.

(d)(i)


Draw a straight, linear line of best fit through the data points on the \(v_{\text{max}}^2\) versus \(m\) graph, trying to keep an equal balance of points above and below your line.

(d)(ii)
To get an experimental value for \(b\), let’s calculate the slope of the best-fit line using two points that lie directly on the line. Let’s use \((0.25\,\text{kg}, 4.5\,\text{m}^2/\text{s}^2)\) and \((0.50\,\text{kg}, 9.0\,\text{m}^2/\text{s}^2)\) as an example:
\( \text{Slope} = \dfrac{9.0 – 4.5}{0.50 – 0.25} = 18\,\text{m}^2/(\text{s}^2\cdot\text{kg}) \)
Since the cylinder is at terminal velocity, acceleration is zero, allowing us to equate the forces:
\( mg – bv_{\text{max}}^2 = 0 \)
\( v_{\text{max}}^2 = \left(\dfrac{g}{b}\right)m \)
This shows that the slope of our graph is equal to \(g/b\). We can rearrange this to solve for \(b\):
\( b = \dfrac{g}{\text{Slope}} \)
\( b = \dfrac{9.8\,\text{m/s}^2}{18\,\text{m}^2/(\text{s}^2\cdot\text{kg})} \approx 0.54\,\text{kg/m} \)

(e)(i)
Vertical axis: Maximum velocity (or \(v_{\text{max}}\))
Horizontal axis: Length of cylinder

(e)(ii)
By plotting the maximum velocity against the different lengths of the cylinders from Set 2 (which all share the same mass), the student can directly observe the relationship. If the graph yields a horizontal line with a slope of zero, it proves that length does not impact the maximum speed. If the graph shows a non-zero slope or a curve, it provides evidence that the cylinder’s length does indeed affect its terminal velocity.

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