Question

If the radial and transverse velocities of a particle are always proportional to each other show that the path is an equiangular spiral

Answer 1

Answer:

The radial and transverse components which they are moving rapidly with an outward direction.  

Where the radial velocity is an object for the path from a fixed point through a straight line.  

And the transverse velocity  which will refers to an object to a path with an angle  θ to the origin path from a fixed point.

radical velocities ∝ transverse velocities

dr/dt ∝ r dθ/dt

dr/dt ∝ λr dθ/dt

Where λ is constant

dr/r = λ dθ

By integrating we get

log r = λθ + log a where log a is constant

Where

  log r - log a = λθ

 log r/a = λθ

 r/a = e^{λθ}

 r = ae^{λθ}

Hence it is an equiangular spiral