Maximizing Horizontal Distance
Date: 04/10/98 at 12:44:47 From: Graham Carter Subject: Algebra/Trigonometry I've been trying to prove that without air resistance, an object needs to be thrown at 45 degrees to obtain the furthest possible horizontal distance. Effectively, I can prove this by stating that the product of sin(A) and cos(A) is greatest when they are the same; ie: 1/sqrt(2). However, I'm finding it difficult to prove this with algebra. Also related to this, I would like to know how to show that if the sum of two numbers is always constant, the product is greatest when they are the same. If you could help, that would be great. Thanks, Graham Carter
Date: 04/10/98 at 14:35:09 From: Doctor Rob Subject: Re: Algebra/Trigonometry To answer your second question first, you want to maximize u*v, subject to the constraint that u + v = c, a constant. That is the same as maximizing u*(c-u) = c^2/4 - (c/2-u)^2 (completing the square!), which is the same as minimizing (c/2-u)^2, and the minimum of this is clearly zero, which happens when u = c/2 = v. The path of the projectile is given by the equations y = -g*t^2/2 + k*sin(A)*t x = k*cos(A)*t Here g is the acceleration of gravity, and k is the object's initial speed. Then y = 0 when t = 0 and when t = 2*k*sin(A)/g, and this is the time of impact, at which time the x-value, that is the range, is: x = 2*k^2*sin(A)*cos(A)/g To maximize this, you need to maximize sin(A)*sqrt[1-sin^2(A)], which is the same as maximizing its square: sin^2(A)*[1-sin^2(A)] = 1/4 - [1/2 - sin^2(A)]^2 This is maximized when [1/2 - sin^2(A)]^2 is minimized. The minimum value of any square is zero. In this case, it means that sin(A) = 1/sqrt(2), so cos(A) = 1/sqrt(2), and the maximum range is 2*k^2*sin(A)*cos(A)/g = k^2/g. Note sin(A) = cos(A) = 1/sqrt(2) means the angle of fire is 45 degrees. This is all done without calculus, which would have helped. A simpler method is to use a trigonometric identity for sin(2*A) to rewrite the range equation as: x = k^2*sin(2*A)/g and this is maximized when sin(2*A) = 1, so 2*A = 90 degrees and A = 45 degrees, and the maximum range is k^2/g. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.