Читать книгу Fundamentals of Numerical Mathematics for Physicists and Engineers - Alvaro Meseguer - Страница 23

A ball is thrown from the lowest point of a hill with initial angle and speed (see the figure below). The height of the hill is given by the expression , where and are measured in meters, , and .

1 Edit a Matlab .m function corresponding to the equation whose solution is the abscissa where the parabola and the hill's profile intersect (for arbitrary initial speed and angle and , respectively).

2 Using Newton's method, find the abscissa of the point A where the ball impacts with the hill. Provide your result with at least five exact figures.

3 With the same initial speed, represent the impact abscissa for initial angles within the range .

4 Find the minimum initial angle that allows the ball to impact beyond , i.e. to the right of the hill's peak. Provide your result with at least three exact figures.

5 Explore the order of convergence of the root‐finding method when computing the minimum angle in (d). Do you observe an increase in the number of iterations required to achieve the desired accuracy? If so, explain what may be the reason for that phenomenon.