300 Week 8: Faraday’s Law and Induction
This is the rate the kinetic energy of the car is beingreducedto either heating the resistor
or recharging the car’s battery. As the kinetic energy is reduced,the car willslow down.
Note well thatω= vr wherer is the radius of the tire. The kinetic energy of the car is
K=1
2
mcv^2 =1
2
mcω^2 r^2.
So here’s the challenge: Convert the expression for the power above into a differential equation
forK, the kinetic energy of the car. Solve for the kinetic energy as a function of time, starting
from an arbitrary initial valueK 0. Note well that the car wouldnever quite stop if only
magnetic braking were used (assuming that the model above is accurate even for very small
speeds). Cars with magnetic brakes mustalwaystransition to friction brakes when the speed
becomes small, because they exert less braking force and remove less energy as the car slows
down!In a car with magnetic brakes the loop would recharge a battery. Inthe next chapter we’ll learn to
treat oscillating voltages and power more accurately, but this estimate should suffice for the moment.