Chapter 9.10, Problem 79E

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

$y=\left(\tan \theta +\frac{g}{k{v}_0\cos \theta }\right)x+\frac{g}{k^2}\ln \left(1-\frac{kx}{v_0\cos \theta }\right)$

where $v_0$ is the initial speed, $\theta$ is the angle of projection, g is die deceleration due to gravity, and k is the drag factor caused by air resistance. Using the power series representation

$\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots ,-1<r<1$

verify tint the trajectory can be rewritten as