Equation for one speed relative to another

If I have a car with a tyre diameter of 50cm and swap the tyres to a smaller diameter of 45cm (therefore affecting the speedometer accuracy), how could I determine an equation that relates the true speed with the display speed?

I'm having trouble getting my head around this and any help is appreciated.

Re: Equation for one speed relative to another

The speedometer works by effectively measuring the angular speed of the wheels. If you put smaller tires on the car, then one revolution of the tires takes the car a shorter distance. Since the linear distance the car travels in one revolution is equal to the circumference of the tire, and the circumference varies linearly as the diameter, how must then the true speed of the car vary with respect to the indicated speed?

Re: Equation for one speed relative to another

I understand that with a smaller tyre circumference the wheel would need to spin faster for the car to go the same speed, which means the speedo counts more rotations and displays a higher speed. But I'm having trouble translating this into an equation in which I can substitute any displayed speed to find the true speed.

Re: Equation for one speed relative to another

Suppose the diameter of the tyres is magically cut in half and the car keeps moving at the same speed...how will this affect the indicated speed?

Re: Equation for one speed relative to another

The indicated speed would be doubled, right? So that would make the equation T = a/b * D, where T=true speed, D=displayed speed, a=diameter of smaller tyre and b=diameter of correct tyre size.

Re: Equation for one speed relative to another

Yes, that's correct! Good work! :D