Transformation of the Alpha Factor Between Inertial Frames

Suppose we have two inertial frames O and O' with O' moving along the x axis of O with a constant speedA particle moving with a velocityin the inertial frame O has a momentumwhereThe same particle in the inertial frame O' has a momentumwhere

In the inertial frame O'(1)

The velocity transformations from O to O' are

Substitution of these expressions into (1) gives

In the inertial frame Oso



Multiplying out the brackets in the numerator and simplifying gives

Now take the square root to give