Problem: Due to a mechanical failure,a ground crew has lost control of a missile recently fired.It is known tha the missile will proceed at a constant speed on a straight course of unknown direction.When the missile is 4 miles away,it is sighted for an instant and lost again.Immediately an anti-missile missile is fired with a constant speed three times that of the first missile.What should be the course of the second missile for it to overtake the first one?
The book's solution is that using the positive x-axis the line from position sighted four miles away to the ground crew.Proceed three miles along this line(in case the missile is returning to base)and then follow the spiral $ r=e^{θ/\sqrt8}$(polar coordinates)(assume both missiles move on the same plane).
I am trying to figure out why we use that spiral,my thoughts are that since $r(θ)=e^{θ/\sqrt8}$ then $log r=θ/\sqrt8$ and $\frac{dr}{dθ}/r=\frac{1}{\sqrt8}$ where r the length of the possition vector of the anti-missile and θ its angle with x-axis.Suppose $0\le \alpha\le π$ the angle between the velocity vector and the position vector of the anti-missile then I know that(from previous exercise) $\cot\alpha=\frac{dr}{dθ}/r$ so i try to prove that $cot\alpha = 1/\sqrt8$ in any possible position of impact. I also think that if $ φ$ the angle between the velocity vectors of the two missiles in a possition of impact then $α=π-φ$. Any hint would be helpfull.Thank you!