Chapter 9.10, Problem 83E

Projectile Motion A projectile fired from the ground follows the trajectory given by

$y=\left(\tan \theta +\frac{g}{k{v}_0\cos \theta }\right)x+\frac{g}{x^2}\ln \left(1-\frac{kx}{v_0\cos \theta }\right)$ where $v_0$ is the initial speed, $\theta$ is the angle of projection, g is the acceleration due to gravity, and k is the drag factor caused by air resistance. Using the power series representation

$\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots ,-1<x<1$ verify that the trajectory can be rewritten as

$y=\left(\tan \theta \right)x-\frac{g{x}^2}{2v_0^2{\cos }^2\theta }-\frac{kg{x}^3}{3v_0^3{\cos }^3\theta }-\frac{k^{2}g{x}^4}{4v_0^4{\cos }^4\theta }-\cdots .$