Variable Radius: Radial Velocity and the Coriolis Term

Apply the product rule to $\vec{r} = r\hat{r}$, substitute $d\hat{r}/dt$, and write $\vec{v} = \frac{dr}{dt}\hat{r} + r\frac{d\theta}{dt}\hat{\theta}$ with the new radial component $v_r = dr/dt$.

Differentiate the velocity term by term and collect $a_r = \frac{d^2r}{dt^2} - r\left(\frac{d\theta}{dt}\right)^2$ and $a_\theta = 2\frac{dr}{dt}\frac{d\theta}{dt} + r\frac{d^2\theta}{dt^2}$, then set $dr/dt = 0$ to recover the circular case.

Name the first tangential term, write it in $\mathrm{m\,s^{-2}}$, and record it as the contribution absent from motion at fixed radius.

Take the four derivatives, assemble $\vec{a} = (2bc - 4bc^3t^4)\hat{r} + 10bc^2t^2\hat{\theta}$, carry the units of $b$ and $c$, and solve for $t_1 = (1/2c^2)^{1/4}$ and $\theta_1 = \sqrt{2}/2$.