The formula to convert from radians per second to kilometers per hour is given by :

$$v_{(km.h^{-1})} =3.6 \times r \times \omega_{(rad.s^{-1})}$$

where :

• $\omega_{(rad.s^{-1})}$ the angular velocity expressed in radians per seconde
• $v_{ (km.h^{-1}) }$ : the linear velocity expressed in kilometers per hour
• $r$ : the radius expressed in meters

By clicking on this link, you will find an online converter from rad/s to km/h

To convert radians per second (rad/s) to kilometers per hour (km/h), you need to consider the relationship between angular velocity (rad/s) and linear velocity (m/s), and then convert the linear velocity from meters per second to kilometers per hour. Here's how to do it:

1. Convert rad/s to m/s:

   • The linear velocity $v$ in meters per second (m/s) is given by $v = \omega \times r$, where $\omega$ is the angular velocity in rad/s and $r$ is the radius of the circular path in meters.

2. Convert m/s to km/h:

   • To convert meters per second to kilometers per hour, multiply the speed in m/s by 3.6 (since 1 m/s is equal to 3.6 km/h).

Let's denote:

• $\omega$ as the angular velocity in rad/s,
• $r$ as the radius of the circular path in meters.

The conversion steps are as follows:

1. Calculate the linear velocity in m/s:
   $$v = \omega \times r$$

2. Convert the linear velocity from m/s to km/h:
   $$v_{\text{km/h}} = v \times 3.6$$

So the combined formula to convert rad/s to km/h is:
$$v_{\text{km/h}} = (\omega \times r) \times 3.6$$

Example Calculation

For example, let's consider a vehicle with wheels of 20 centimetres radius rotating at a speed of 5 rad/s, the speed of the vehicle in km/h is given by :

1. Calculate the linear velocity in m/s:
   $$v = 5 \times 0.2 = 1 \text{ m/s}$$

2. Convert the linear velocity from m/s to km/h:
   $$v_{\text{km/h}} = 1 \times 3.6 = 3.6 \text{ km/h}$$

Thus, 20-centimetre wheels turning at 5 rad/s correspond to a displacement of 3.6 km/h.