Chapter MPS, Problem 14.6MPS

To determine

Use the double-angle identity to simplify the equation.

Answer to Problem 14.6MPS

  $x=\frac{2}{g\left(v^2\sin 2\theta \right)}$

Explanation of Solution

Given information:

Scott was solving a series of equations concerning maximizing the distance for throwing a rock off a bridge when he came across the equation.

  $x=\frac{1}{g\left({\upsilon }^2\sin \theta \cos \theta \right)}$

Calculation:

Consider the following equation;

  $x=\frac{1}{g\left({\upsilon }^2\sin \theta \cos \theta \right)}$

Recollect the double-angle identity,

  $\sin 2\theta =2\sin \theta \cos \theta$

Simplify the given equation as follows;

  $\begin{array}{l}x=\frac{1}{g\left({\upsilon }^2\sin \theta \cos \theta \right)}\\ =\frac{1}{g\left(v^2\frac{\sin 2\theta }{2}\right)}\\ =\frac{2}{g\left(v^2\sin 2\theta \right)}\end{array}$

Hence, the simplified form of the given equation is $x=\frac{2}{g\left(v^2\sin 2\theta \right)}$ .