What is the distance travelled by the particle in polar coordinates?

Question

Suppose a particle moves in a path $r=f(\theta)$ with constant velocity. $s$ is a distance travelled by the particle from $\theta=0$ to a point on the curve. Actually I want an expression for $\frac{ds}{d\theta}$. In the notes it is given as $\frac{ds}{d\theta}=r$. I assume they got this from the equation $s=\int_0^\theta r d\theta$. Is this correct? We should use arc-length here? Please help me.

Answer 1

The arc length of a curve in polar form, between angles $\theta_1$ and $\theta_2$, is given by:

$$\int_{\theta_1}^{\theta_2}\sqrt{(r'(\theta))^2+(r(\theta))^2}\,d\theta$$

And if by "constant velocity" you mean $r'(\theta)$ is constant, then the notes are incorrect. It is only when $r$ itself is constant that $\frac{ds}{d\theta}=r$ as is seen by inspection of the above formula ($r'(\theta)=0$).