Associated Topics || Dr. Math Home || Search Dr. Math

### 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

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/
```
Associated Topics:
High School Basic Algebra
High School Physics/Chemistry
High School Trigonometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search