A conductor bar of mass m is thrown to the right with velocity $\vec {v} _0 $ over a pair of parallel and metallic rails separated by a distance L, as illustrated in the figure below. The rails are supported in a horizontal plane, have their left ends connected by a resistor resistor R and are in the presence of uniform and pointing upward vertical magnetic field. Whereas the bar and rails are ideal conductors and the bar slides without friction,
(a) determine the rate of change of the kinetic energy of the bar;
(b) plot the kinetic energy of the bar as a function of time.
How to solve without Calculus?
My solution: 
Just use conservation of energy. The power loss in the resistor is equal to the rate of change in kinetic energy. $$\varepsilon=-BLv\\P=\frac{\varepsilon^2}R=\frac{B^2L^2v^2}R$$ Then $$P=-\frac{dE_K}{dt}=\frac{B^2L^2v^2}R=\frac{2E_KB^2L^2}{Rm}$$ At this point you would need to show that this is a logarithmic curve