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

To convert angular velocity from radians per second (rad/s) to linear velocity in kilometers per hour (km/h), you need to consider the radius of the circular path. Here's the general process:

### Step 1: Understand the relationship between angular and linear velocity

The linear velocity $v$ (in meters per second, m/s) can be calculated from the angular velocity $\omega$ (in radians per second, rad/s) using the following relationship:

$v = \omega \times r$

where:

- $v$ is the linear velocity in m/s.
- $\omega$ is the angular velocity in rad/s.
- $r$ is the radius of the circular path in meters.

### Step 2: Convert linear velocity from m/s to km/h

After obtaining the linear velocity in m/s, you can convert it to km/h using the following conversion factor:

$v_{\text{km/h}} = v_{\text{m/s}} \times \frac{3600 \, \text{s/h}}{1000 \, \text{m/km}} = v_{\text{m/s}} \times 3.6$

### Step 3: Convert angular velocity in rad/s to linear velocity in km/h

By merging step 1 and 2, we can get the following relation from rad/s to

$v_{\text{m/s}} = \omega_{\text{rad/s}} \times r$

$v_{\text{km/h}} = \omega_{\text{rad/s}} \times 3.6 \times r$

We finally get:

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

### Example Conversion

Let's say you have an angular velocity of $\omega = 10 \, \text{rad/s}$ and a radius $r = 2 \, \text{meters}$.

**Calculate the linear velocity in m/s:**

$v_{\text{m/s}} = \omega \times r = 10 \, \text{rad/s} \times 2 \, \text{m} = 20 \, \text{m/s}$

**Convert the linear velocity from m/s to km/h:**

$v_{\text{km/h}} = 20 \, \text{m/s} \times 3.6 = 72 \, \text{km/h}$

### Summary

To convert rad/s to km/h, first multiply the angular velocity by the radius (in meters) to get the linear velocity in m/s. Then, multiply that result by 3.6 to convert it to km/h.