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(63)

where
stands for scalar product of the vectors f and dl.
Applying equation (56) in case when force is acting on the object X in direction of the speed vx/m , we have

(64)

We may substitute in equation (64):
Thus
(65)

The equation (65) determines kinetic energy of an object X in its own frame, with reference to a M-frame, moving at a speed vm/x with reference to an object X.

Common equation for kinetic energy in frame M can be obtained by substitition for the force from equation (57) to equation (63) and by solving of respective integral. In case when acceleration is in line with the speed, applying equations (57a), (57b), and (63), we obtain :

(66)

Remark: We can derive :
Common equation for galileian projection of the kinetic energy from M- to N-frame can be obtained by substitution for the force from equation (58) to equation (63).Under the circumstances that M-frame moves without acceleration and without rotation with reference to N-frame, and that acceleration of the object X in M-frame is in line with the speed vx/m , we may substitute for the force in equation (63) from equation (58b), and we have:

(67)

There are two con-spacetime accelerating mechanisms bringing an object to the same speed with reference to any reference frame:
• By using the means of the object own reference frame. Energy required to reach the speed is lower (see equation 65).
• By using the means of any inertial reference frame . Energy required to reach the speed is higher now (see equation 66). Braking the object X that has been accelerated to a speed vx/m using the means of the object own reference frame, in a reference frame M, brings gain of kinetic energy.

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