Convert revolutions per minute [rpm] to kilometers per hour [km/h] with gear ratio and vice-versa

Online converter

Convert any value from / to revolutions per minute [rpm] to kilometers per hour [km/h], angular velocity to linear velocity with gear ratio. Fill one of the following fields, values will be converted and updated automatically.

rpm
ratio (1:x)
km/h

Explanation

First let's consider the gear ratio. The ratio divides the input angular velocity according to the following formula:

$$N^{output}_{(rpm)}= \dfrac { N^{input}_{(rpm)}}{ratio}$$

The angular velocity of the wheel (in rpm) can be converted into $$m.s^{-1}$$ thanks to the following formula:

$$v_{(m.s^{-1})} = r \times \omega_{(rad.s^{-1})} = r \times \frac {2 \pi }{60.ratio}.N^{input}_{(rpm)}$$

It becomes easy to convert this linear velocity expressed in $$m.s^{-1}$$ into $$km.h^{-1}$$ thanks to the following formula:

$$v_{ (km.h^{-1}) } = \frac {3600}{1000}.v_{(m.s^{-1})} = \dfrac{3.6}{ratio} \times v_{(m.s^{-1})}$$

By merging the two previous equations, we can deduce the relationship between revolution per minute [rpm] and kilometers per hour [km/h]:

$$v_{ (km.h^{-1}) } = \frac {3600}{1000} \times r \times \frac {2 \pi }{60.ratio}.N_{(rpm)} = \frac {3}{25.ratio}. \pi. r . N^{input}_{(rpm)}$$

et vice versa:

$$N^{input}_{(rpm)} = \frac {25.ratio} { 3. \pi . r} v_{(km.h^{-1})}$$