Variation of the dog chasing a cat problem

[timetraveller123]

Homework Statement

Homework Equations

The Attempt at a Solution

this is in the cats frame

let the radial length be r then

$v cos\theta = v _{\theta}\\ v - vsin \theta = v _r \\ \frac{dr}{dt} = \frac{dr}{d \theta} \frac{d \theta}{dt} = v - vsin \theta = \frac{dr}{d \theta} \frac{v cos \theta}{r}\\ \frac{dr}{r} = \frac{1 - sin \theta}{cos \theta} d\theta\\ \int _{L} ^{r} \frac{dr}{r} = \int _{0}^{\frac{\pi}{2}}\frac{1 - sin \theta}{cos \theta} d\theta\\$
hopefully this is correct
doing the right integral got me ln2
so it become

$ln r - ln L = ln 2\\ r = 2L\\$
but the answer was r = L/2 what am i doing wrong

 

[mfb]

Your minimal distance is twice the initial distance?
Looks like a simple sign error. Your dr/dt is positive, but the actual distance is decreasing.

 

[timetraveller123]

ah yes thanks so careless :)