In polar coordinates, the position vector is

$$\mathbf r=r\hat{\mathbf r}.$$

Both the radial coordinate $r(t)$ and the radial unit vector can change with time. Differentiating therefore requires the product rule and the time derivatives of the moving polar basis.

Velocity

Differentiate the position vector:

$$\mathbf v =\frac{d}{dt}(r\hat{\mathbf r}) =\dot r\hat{\mathbf r}+r\frac{d\hat{\mathbf r}}{dt}.$$

Since

$$\frac{d\hat{\mathbf r}}{dt}=\dot\theta\hat{\boldsymbol\theta},$$

we obtain

$$\boxed{\mathbf v =\dot r\hat{\mathbf r}+r\dot\theta\hat{\boldsymbol\theta}}.$$

The first term is the radial velocity. The second is the transverse or azimuthal velocity.

The speed satisfies

$$\boxed{v^2=\dot r^2+r^2\dot\theta^2}$$

because the two polar unit vectors are perpendicular.

Acceleration

Differentiate velocity:

$$\mathbf a =\frac{d}{dt}\left(\dot r\hat{\mathbf r}+r\dot\theta\hat{\boldsymbol\theta}\right).$$

For the radial term,

$$\frac{d}{dt}(\dot r\hat{\mathbf r}) =\ddot r\hat{\mathbf r} +\dot r\dot\theta\hat{\boldsymbol\theta}.$$

For the transverse term,

$$\frac{d}{dt}(r\dot\theta\hat{\boldsymbol\theta}) =(\dot r\dot\theta+r\ddot\theta)\hat{\boldsymbol\theta} +r\dot\theta\frac{d\hat{\boldsymbol\theta}}{dt}.$$

Using

$$\frac{d\hat{\boldsymbol\theta}}{dt}=-\dot\theta\hat{\mathbf r},$$

and collecting radial and transverse components gives

$$\boxed{\mathbf a =(\ddot r-r\dot\theta^2)\hat{\mathbf r} +(r\ddot\theta+2\dot r\dot\theta)\hat{\boldsymbol\theta}}.$$

Each term has a distinct geometric origin.

Radial acceleration

The radial component is

$$\boxed{a_r=\ddot r-r\dot\theta^2}.$$

The term $\ddot r$ describes change in radial speed. The term

$$-r\dot\theta^2$$

points inward and remains even when $r$ is constant. For circular motion with constant radius,

$$a_r=-r\omega^2=-\frac{v^2}{r},$$

which is the familiar centripetal acceleration.

Transverse acceleration

The transverse component is

$$\boxed{a_\theta=r\ddot\theta+2\dot r\dot\theta}.$$

The term $r\ddot\theta$ comes from changing angular speed. The term

$$2\dot r\dot\theta$$

appears when radial motion and angular motion occur simultaneously: changing radius changes the transverse velocity even if $\dot\theta$ itself is momentarily constant.

Worked example

Suppose at one instant a particle has

$$r=2.0,\mathrm m,$$

$$\dot r=1.0,\mathrm{m/s},$$

$$\ddot r=0,$$

$$\dot\theta=3.0,\mathrm{rad/s},$$

and

$$\ddot\theta=2.0,\mathrm{rad/s^2}.$$

Its velocity components are

$$v_r=\dot r=1.0,\mathrm{m/s},$$

$$v_\theta=r\dot\theta=(2.0)(3.0)=6.0,\mathrm{m/s}.$$

Thus

$$\mathbf v=1.0\hat{\mathbf r}+6.0\hat{\boldsymbol\theta},\mathrm{m/s}.$$

The radial acceleration is

$$a_r=0-(2.0)(3.0)^2=-18,\mathrm{m/s^2},$$

and the transverse acceleration is

$$a_\theta=(2.0)(2.0)+2(1.0)(3.0)=10,\mathrm{m/s^2}.$$

Therefore

$$\boxed{\mathbf a=-18\hat{\mathbf r}+10\hat{\boldsymbol\theta},\mathrm{m/s^2}}.$$

The large inward acceleration is produced mainly by the rapid change in direction of the moving polar basis.

Polar kinematics is useful whenever both radius and angle evolve, including spiral motion, central-force mechanics, orbital motion, rotating mechanisms, and many particle-dynamics problems.