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Illustration Orbital velocities and angles in a circular motion

Orbital velocities and angles for circular motion
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Two velocity vectors \(\boldsymbol{v}_1\) and \(\boldsymbol{v}_2\) at two different times of a body moving on a circular path. The vector (\(\boldsymbol{v}_2\) was parallel shifted here. The magnitude of the two vectors is equal: \(v_1 = v_2\). The direction, on the other hand, has changed due to the motion on a curved circular path. This change of direction results in the change \(\Delta \boldsymbol{v}\) of the velocity vector:$$ \Delta \boldsymbol{v} ~=~ \boldsymbol{v}_2 - \boldsymbol{v}_1 $$

The two right triangles (green and blue) with the corresponding angles were used to derive the centripetal acceleration. It can be shown that \(\Delta \varphi = \Delta \theta \).

The velocity change \(\delta \boldsymbol{v}\) is the cause of the centripetal acceleration acting on a body moving in a circular orbit.