Question #43932

A block of mass $m = 1 \text{ kg}$ is released from rest and slides down a rough inclined plane of angle $\theta = 30^\circ$. A small generator attached to the block produces an electrical power $P = c v^2$, where $v$ is the instantaneous velocity of the block and $c = 0.2 \text{ W}\cdot\text{s}^2/\text{m}^2$. This power is entirely dissipated as heat by a coil embedded in the bottom surface of the block. The coefficient of kinetic friction between the block and the incline decreases with the dissipated power as $\mu_k = \mu_0 - k P$, where $\mu_0 = \frac{1}{2\sqrt{3}}$ and $k = \frac{1}{20\sqrt{3}} \text{ W}^{-1}$. Assuming the thermal response is instantaneous, what is the kinetic energy of the block after it has slid a distance $d = 10 \text{ m}$ down the incline? (Take $g = 10 \text{ m/s}^2$)