Addition of Velocities Notes for Engineering Physics BTech 1st Year
Addition of Velocities Notes for Engineering Physics:
Addition of velocities is simply the addition of vectors.
direction of the vector and its components on the x and y axis of the coordinate system:

Magnitude : v=√v²river+v²boat

Direction : θ=tan−1(v river * v boat)

xcomponent : v river=vcosθ

ycomponent : v boat=vsinθ
The first two equations are for when the magnitude and direction are known, but you are looking for the components. The last two equations are for when the components are known, and you are looking for the magnitude and direction. The magnitude of the observed velocity from the shore is the square root sum of the squared velocity of the boat and the squared velocity of the river.
Einstein Velocity Addition:
The Einstein velocity relationship transforms a measured velocity as seen in one inertial frame of reference (u) to the velocity as measured in a frame moving at velocity v with respect to it (u’). In problems involving more than two objects, the main difficulty is the assignment of velocity to all the objects
If A sees B moving at velocity v, then a velocity measured by B (u’) would be seen by A as:
Courtesy : http://hyperphysics.phyastr.gsu.edu