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#### 5.3.2.2 Speed and acceleration of the free object in a spaceflow line

We can judge from Tab1 that for gravitational objects up to 1033 kg due to small value of the ratio vm/x/c the relativistic factors may be neglected for practical calculations (perhaps except of the calculations of the long-term gravitational influence).
Differentiating the equations (179a) and (179b), we have :
(181)

(181a)

Substituting from equations (180) and (181) to equation (171b), and neglecting the relativistic factors, we receive:
(181b)

where
gzn/m-
stands for the Zct acceleration of the object on its downward motion
And, substituting in a similar way from equations (180a) and (181a) to equation (171b), we receive :
(181c)

where
gzn/m+
stands for the Zct acceleration of the object on its upward motion
Now, designating :
(181d)

we can derive :
(181e)

(181f)

Applying c=2,997*108 m/s and the data from Tab1, we are able to calculate that even for the gravitational object of the mass of 2*1033 kg the ratios g2+/1 and g2-/1 do not exceed the value of 5,1 %. This is why we can say, that for practical calculations (except of the calculations of the long-term gravitational influence) the equations (181b) and (181c) may be modified, and the Zct object acceleration defined as :
(182)

Neglecting the relativistic factors and applying the equation (182), we can rewrite the equation (174):
(183)

Resolving the differential equation (183) for the boundary condition vzn/m=vzn/mk at rlmx=rlmxk, we receive the common formula for the Zct speed of the object moving in a spaceflow line:
(184)

Applying the equation (108) for the speeds:
vzn/m
the object's Zct speed (speed of the object in length unit of the space density that the object is just passing through, within time unit of the chosen frame),
vn/m
the object's frame intrinsic speed (speed of the object in length and time units of the space density that the object is just passing through), and,
von/m
the object's Zcr speed (object,s frame intrinsic speed, expressed in length and time units of the chosen frame),
we can write:
(185)

(185a)

Since, neglecting the relativity factor,
(185b)

substituting from the equations (184) and (185b) to the equation (185) we obtain:
(186)

where
vn/mk
stands for the object's frame intrinsic speed, corresponding to its Zct speed vzn/mk, at the radius rlmxk (the boundary condition).
(186a)
and, substituting from equations (186) and (185b) to equation (185a), we obtain:
(187)

where
von/mk
stands for the object's Zcr speed, corresponding to its frame intrinsic speed vn/mk, and to its Zct speed vzn/mk, at the radius rlmxk (the boundary condition).
(187a)
The equations (186) and (187) make possible to derive the respective frame intrinsic and Zcr acceleration, applying the following equations:
(188)

(188a)

Resolving the equations (186) and (188) we obtain :
(189)

Remark 1 :
Once the object has reached the frame intrinsic speed vn/mk=-vm/x at any radius rlmxk it moves in its following course without its relative motion with respect to the spacetime structure. Therefore it must be gn/m=gx/m=0. We can see, that in this case the equation (189) gives gn/m=0. This result is in accordance with the equation (133a).
Once the object has reached the frame intrinsic speed vn/mk=vm/x (in a drection out of the gravitational field), its frame intrinsic acceleration again becomes gn/m=0 (see equation 189), and the speed vn/m=vm/x will not come back below the limit vm/x without being braked by the outward force (see equation (186). The speed vm/x therefore represents the frame intrinsic escaping speed from the gravitational field.
Resolving the equations (187) and (188a) we have :
(190)

Remark 2 :
The object moving at the frame intrinsic speed vn/mk=vm/x (without motion with respect to the spacetime structure), that is, at the Zcr speed
(190a)

does not appear to move without acceleration in Zcr frame. In fact in this frame it appears to be braked, since in this case the equation (190) gives the acceleration
(190b)

This result is in accordance with the equation (137).
Remark 3 :
Resolving the equation (190) for gon/m=0, we receive:
(190c)

This equation defines radius rlmx at which the Zcr acceleration, or braking of the object in free fall reaches zero, that is, the radius at which the object's Zcr accelelation or braking in course of its free fall has been reduced to zero and the object now begins to be braked or accelerated.
Remark 4 :
Substituting for vn/mk=0 to equation (189), we receive:
(190d)

The equation (190d) says, that the frame intrinsic acceleration of the object not moving with respect to M-frames (or, with respect to the gravitational object) is constant at rlmx = rlmxk (that is at the moment when vn/m = 0), and then becomes decreasing with decreasing radius (since vn/m goes up).
Remark 5 :
Substituting for von/mk=0 to equation (190), we receive the equation
(190e)

defining the course of the Zcr acceleration of the object along with radius rlmxk, the speed of that has become zero at the radius rlmxk (at the radius rlmxk the object has not been moving with respect to M-frames).
The examples of the object free fall acceleration calculated in accordance to the equations (182), (188) and (188a) are shown on Fig13 and Fig14.

Fig13

Fig14

Resolving the equation (186) for vn/m=0, we can derive the radius rc at which the object culminates if its motion has begun at the radius rlmxk and at the speed vn/mk of a positive value:
(191)

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