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4.12 Kinetic energy (w)

image
image (63)

where
image
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
image
image (64)

We may substitute in equation (64):
am/x
acc. to equation (56b),
dtm/x
image
and the following solution:
Since
image
for the motion at a constant acceleration ax/m in frame M it must be:
image
Thus
image (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 :
image
image (66)


Remark: We can derive :
image
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:
image
image (67)


There are two con-spacetime accelerating mechanisms bringing an object to the same speed with reference to any reference frame:

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