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?

I noticed no one proved part 1.  Was it too trivial?  I guess it is rather trivial, but part 2 is beyond my math knowledge.

The distance the ball travels before landing is directly proportional to the product of the sine and cosine of the angle.  To understand this, imagine the initial velocity of the ball as having two components, vertical and horizontal (horiz=cos, vert=sin).  The horizontal is equal to the velocity of the ball.  The vertical is proportional to the time spent in the air.  dist=rate x time.

So I must optimize this value:
sinθcosθ
=1/2 sin(2θ)
At maximum, 2θ=pi/2
θ=pi/4

I wonder if I could figure out a way to do part 2 even with my lack of calc skills.

Posted by Tristan on 2005-03-19 02:51:46