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Make it go! (Posted on 2005-03-18) Difficulty: 4 of 5
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?

See The Solution Submitted by Federico Kereki    
Rating: 4.5000 (2 votes)

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Proof that the second part needs a numberical solution | Comment 8 of 17 |
Charlie has stated how to set up the problem. The formula for the lenght of the trajectory I get by doing the integral is,

L = K * (sin a + (cos a)^2*ln [tan a + sec a])

Where K = v^2/g, and a is the kicking angle.

To maximize L with respect to the kicking angle,

dL/da ~ 1 - sin a * ln[tan a + sec a] = 0

This is a trancendental equation that can only be solved numerically. There is no need to do that at this point because Charlie's answer of 56.5 works straightaway.

Edited on March 19, 2005, 2:49 pm

Edited on March 22, 2005, 3:55 am
  Posted by ajosin on 2005-03-19 14:48:30

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