Linear Momentum AP Physics C Mechanics FRQ – Exam Style Questions etc.
Linear Momentum AP Physics C Mechanics FRQ
Unit 4: Linear Momentum
Weightage : 15-25%
Question
In an experiment, a student wishes to use a spring to accelerate a cart along a horizontal, level track. The spring is attached to the left end of the track, as shown in the figure above, and produces a nonlinear restoring force of magnitude Fs = As2 + Bs, where s is the distance the spring is compressed, in meters. A measuring tape, marked in centimeters, is attached to the side of the track. The student places five photogates on the track at the locations shown.
(a) Derive an expression for the potential energy U as a function of the compression s. Express your answer in terms of A, B, s, and fundamental constants, as appropriate.
In a preliminary experiment, the student pushes the cart of mass 0.30 kg into the spring, compressing the spring 0.040 m. For this spring, A = 200 N/m2 and B = 150 N/m. The cart is released from rest. Assume friction and air resistance are negligible only during the short time interval when the spring is accelerating the cart.
(b) Calculate the following:
i. The speed of the cart immediately after it loses contact with the spring
ii. The impulse given to the cart by the spring
In a second experiment, the student collects data using the photogates. Each photogate measures the speed of the cart as it passes through the gate. The student calculates a spring compression that should give the cart a speed of 0.320 m s after the cart loses contact with the spring. The student runs the experiment by pushing the cart into the spring, compressing the spring the calculated distance, and releasing the cart. The speeds are measured with a precision of ±0.002 m s. The positions are measured with a precision of ±0.005 m.
| Photogate | 1 | 2 | 3 | 4 | 5 |
| Cart speed (m/s) | 0.412 | 0.407 | 0.399 | 0.374 | 0.338 |
| Photogate position (m) | 0.20 | 0.40 | 0.60 | 0.80 | 1.00 |
(c) On the axes below, plot the data points for the speed v of the cart as a function of position x. Clearly scale and label all axes, as appropriate.
(d)
i. Compare the speed of the cart measured by photogate 1 to the predicted value of the speed of the cart just after it loses contact with the spring. List a physical source of error that could account for the difference.
ii. From the measured speed values of the cart as it rolls down the track, give a physical explanation for any trend you observe.
Answer/Explanation
Ans:
(a)
v = – Ws
\(v = – \int_{0}^{s}\left ( AS^{2}+BS \right )ds\)
\(v = – \left [ \frac{AS^{3}}{3}+\frac{BS^{2}}{2} \right ]\)
(b) i.
\(\frac{(200)(.040)^{3}}{3}+\frac{(150)(.040)^{2}}{2}=\frac{1}{2}(-3)v^{2}\)
\(v = .91, \frac{M}{S}\)
ii.
J = MΔV
= .30 (.91)
J = .273
(c)
(d) i.
\(.320\frac{M}{S} \) \(.412\frac{M}{S} \)
The student pushes the court back too far causing the resulting impulse from the spring to be greater.
ii.
a frictional force causes an acceleration that opposes the carts vel. causing it to slow down.