You kick a ball over a flat field. Taking into account gravity, but disregarding everything else like wind, friction, bounces, etc., etc., at what angle should you kick it so the ball lands the farthest away from you? And at what angle should you kick it so the ball makes the longest trajectory before landing?

(In reply to re: Proof that the second part needs a numberical solution by Charlie)

A plot of

dL/da ~ 1 - sin a * ln[tan a + sec a]  = f(a)

reveals that there are two solutions for f(a) = 0, one at -56.5 and another one at 56.5. In fact, the plot indicates that f(a) is a symmetric function. It was not obvious to me at first, but it can be proved that f(a) = f(-a).

That being said, I do not know why the iterative method Charlie uses to solve the ecuation only converges to the solution at -56.5. The program is still usefull; since f(a) = f(-a) we know that if -56.5 is a solution, then 56.5 must also be a solution (even though we can't get to it).

On the other hand, if one uses Newton's iterative formula,

a = a_0 - f(a_0)/f'(a_0)

then the convergence is better behaved, although the program becomes more complicated because of the new f'(a_0) term.

Posted by ajosin on 2005-03-20 04:04:34