Alvaro Meseguer - Fundamentals of Numerical Mathematics for Physicists and Engineers

13 13. (A–N) Dynamics and Kinematics: A trapeze artist is at rest on top of a platform of height (hollow circle A in the following figure) holding a rigid rod of length and negligible mass that can rotate around the fixed point O. The performer starts falling under the effect of gravity (dashed gray curve), passing through point at maximum speed without touching the ground. Later, the acrobat releases the rod when the angle it forms with the vertical axis is (point B). From point B onward, the artist traces a parabolic trajectory, finally touching the ground at point C, whose abscissa is . Neglect friction forces.Show that .Find the angle for which .What is the angle so that the acrobat maximizes the range ?Provide your numerical answers with at least four exact figures. Advice: for parts (b) and (c), consider the alternative equation

1 1First obtained in the mid‐sixteenth century by the Italian mathematician Scipione del Ferro, inspired by the work of Lodovico Ferrari, a disciple of Gerolamo Cardano.

2 2First derived by the German astronomer Johannes Kepler in 1609.

3 3See Appendix H of Eisberg and Resnick (1985).

4 4National Institute of Standards and Technology.

5 5The etymology of the word is the name of the Persian mathematician and astronomer Al‐Khwārizmī‐ from the eighth century.

6 6is said to be of Cauchy‐type if , such that .

7 7Named after the English mathematicians Sir Isaac Newton (1642–1727) and Joseph Raphson (1648–1715).

8 8This will be addressed in the second part of the book.

9 9This approximation is crucial in Newton's method for systems of equations (Part II).

10 10However, some texts consider the bisection method to be linearly convergent if one considers an upper bound of the error , so that .

11 11The asymptotic constants of secant or chord methods have the same problem with double roots (see Exercises 1.3 and 1.4).

12 12Also known as the false position method (see Exercise 1.6).

13 13Edmond Halley, English astronomer who predicted in 1705 the orbital periodicity of the comet that now bears his name.

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