You use a standard 12V DC motor to power the drive wheel on

You use a standard 12V DC motor to power the drive wheel on a toy car. The nominal rotational velocity of the motor when driven at 12V is 8000 RPM. The car has a 50mm diameter wheel that is being driven. What duty cycle (100% is full on, 0% is completely off) do you need to drive the motor with in order to make the car travel at 15 m/s?

Solution

V=12 V and speed N=8000 RPM

Car Velocity u=15 m/s

Diameter of wheel D=50 mm

radius R=D/2=50/2=25 mm=25*10^-3 m( meters)

Linear velocity u=Raidus*angular velocity W

u=R*W gives W=u/R

Angular Velocity W=u/R=15/(25*10^-3)=600 rad/sec

W=600 rad/sec

New speed in RPM N1=W*60/(2*PI)=600*60/(2*3.14)=5729.28 RPM

Voltage V proportional to speed N

so V1/V=N1/N that gives V1=V*N1/N

Voltage needed for car velocity(15 m/s) is V1=12*5729.28/8000=8.59 V

V1=8.59 V

Duty Cycle=V1/V=8.59/12=0.716

Duty Cycle=0.716 i.e 71.6%