Polar Co-ordinates Homework: Boat Docking w/ Ship

[orochimaru]

Homework Statement

A boat is to dock with a ship. The ship sails along a straight course with speed v. The boat moves with constant speed nv, its motion always being always directed towards the ship. Show that the polar equation of the course of the boat as observed from the ship is

$$\frac{A}{r} = sin\theta tan^n\frac{\theta}{2}$$

where a is a constant and the origin of co-ordinates is the ship and the x-axis is in the direction of the ship's motion

Homework Equations

$$\frac{\overrightarrow dr}{dt} = \frac{dr}{dt}\hat{r} + r\frac{d\theta}{dt}\hat{\theta}$$

The Attempt at a Solution

I introduce a velocity v in the negative for both the boat and the ship in the negative x direction so the ship remains at the origin and made an expression for the components of the boats velocity.

$$\ (-nv -vcos\theta) \hat{r} \\ - v sin\theta \hat{\theta}$$

so $$\frac{dr}{dt} = - nv - vcos\theta \\ \frac{d\theta}{dt} = \frac{v sin\theta}{r}$$

I tried to use the chain rule $$\frac{dr}{dt}\frac{dt}{d\theta}$$ to get $$\frac{dr}{d\theta}$$

and integrate to get r but I got stuck and I don't understand where the constant A comes from.

Any help much appreciated

 

[cristo]

and integrate to get r but I got stuck and I don't understand where the constant A comes from.

Well, I can't see what you've done, and therefore cannot tell you where you're getting stuck. Isn't A just an integration constant?