The classic brachistrochrone problem is standard material in intermediate mechanics. Many variations exist including some accessible to introductory students. While a quantitative solution isn’t feasible in introductory classes, qualitative discussions can be very beneficial since kinematics, Newton’s laws, energy conservation, and motion along curved trajectories all play a role. In this work, we describe an activity focusing on a qualitative understanding of the brachistochrone and examine the performance of freshmen, juniors, and graduate students. The activity can be downloaded at https://w3.physics.arizona.edu/undergrad/teaching-resources.
References
1.
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Masses released from lower points have a smaller oscillation amplitude. For example, if yinitial = 1 m, ay ranges from .
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While we believe these graduate students wrote their solutions quickly, this is disappointing.
13.
One can argue about how to count here. The required acceleration changes as does the angle between n and mg. However, since the curvature and the speed both (independently) affect the acceleration, we consider this to be three reasons.
14.
One can contrast this with the maximum tension of 3mg when a simple pendulum is released from rest from the horizontal. The pendulum requires twice the radial acceleration at its lowest point since a circle’s radius of curvature is half as large as the corresponding cycloid’s.
15.
While their instructor considered this a weak class, the author has seen juniors perform poorly on these qualitative questions multiple times.
© 2021 Author(s). Published under an exclusive license by American Association of Physics Teachers.
2021
Author(s)
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