Physics

This problem combines polar coordinates with kinematics—you need to find how fast the particle is moving along the spiral $r = 2\theta$. The key insight is that speed in polar coordinates isn't just $\frac{dr}{dt}$; you must account for both the radial and tangential components of velocity. Use the formula $v = \sqrt{\left(\frac{dr}{dt}\right)^2 + \left(r\frac{d\theta}{dt}\right)^2}$, then substitute $\theta = t$ to evaluate everything at $t = \pi$.

This problem combines polar coordinates with kinematics—you need to find how fast the particle is moving along the spiral $r = 2\theta$. The key insight is that speed in polar coordinates isn't just $\frac{dr}{dt}$; you must account for both the radial and tangential components of velocity. Use the formula $v = \sqrt{\left(\frac{dr}{dt}\right)^2 + \left(r\frac{d\theta}{dt}\right)^2}$, then substitute $\theta = t$ to evaluate everything at $t = \pi$.