# Work, energy, power?

A body of mass $m$ slides down a plane inclined at an angle $\alpha$. The coefficient of friction is $\mu$. Find rate at which kinetic plus gravitational potential energy is dissipated at any time $t$.
Motion exists along the surface of the plane only (call this direction horizontal). Analyse the weight force $B=mg$ in two components: the vertical and horizontal forces, $N=mg\cos(a)$ and $F=mg\sin(a)$ respectively. Along the horizontal direction, two forces are applied: $F$ and the friction $T=\mu m g \cos(a)$. Call $v(t)$ the velocity at time $t$. By Newton's law, $F-T=mv'(t)$, and solve to get $v(t)=g(\sin(a)-\mu \cos(a))t$ (assume the body was resting at $t=0$).
If $h(t)$ is the height and $S(t)=\int_0^t v(s)ds$ the length traveled at time $t$, then a simple argument shows $h(t)=S(t)\sin(a)$. Now, the total energy is $E(t)=mgh(t)+1/2mv^2(t)=mgS(t)\sin(a)+1/2mv^2(t)$. Differentiate to get $E'(t)=mgv(t)\sin(a)+mv(t)v'(t)$...