In a recent submission to The Physics Teacher, we related how trigonometric identities can be used to find the extremes of several functions in order to solve some standard physics problems that would usually be considered to require calculus. In this work, the functions to be examined are polynomials, which suggests the utilization of the multiple angle identities. Please note that an arrow indicates a step in which quantities irrelevant to the extremization process are simply dropped; as examples, adding a constant term to a function doesn’t change the domain positions of its extremes, nor does rescaling it by a positive factor. Rescaling by a negative factor of course exchanges its maximums and minimums.
References
1.
D.
Baum
, “Solving some ‘calculus-based’ physics problems with trigonometry
,” Phys. Teach.
57
, 470
(Oct.
2019
).2.
John S.
Thomsen
, “Maxima and minima without calculus
,” Am. J. Phys.
52
, 881
(Oct.
1984
). In the solution to the original problem, the potential energy expression is quadratic and the minimum is found by forcing the equation’s discriminant to be zero.3.
One might argue that h can be found more quickly using Archimedes’ principle, which is the equivalent of having the forces sum to zero, which in turn corresponds to dU/dh = 0. Archimedes’ principle does not, however, determine whether the equilibrium is stable or unstable. The current method, however, immediately shows that the position h is a stable equilibrium, since it finds directly the minimum of U(h). Even a calculus approach would require evaluating the second derivative to demonstrate stability.
4.
P.
Palffy-Muhoray
and D.
Balzarini
, “Maximizing the range of the shot put without calculus
,” Am. J. Phys.
50
, 181
(Feb.
1982
). The problem discussed is mathematically identical to that in the present work and was solved using vector algebra.5.
S. K.
Bose
, “Maximizing the range of the shot put without calculus
,” Am. J. Phys.
51
, 458
(May
1983
). The problem discussed is mathematically identical to that in the present work and was solved by forcing the discriminant of the projectile’s path equation to be zero.6.
To see this, solve for the largest angle that will deliver the projectile to the foot of the wall by setting h = 0 in Eq. (3). An object launched at an angle greater than this value will fall short.
© 2020 American Association of Physics Teachers.
2020
American Association of Physics Teachers
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