The Force is closer...NOT!

The centripetal force is given by  \({ F }_{ ob }={ g }_{ m }(e^{ (\cfrac { x-Orbs }{ { r }_{ m } } ) }-e^{ -\cfrac { Orbs }{ { r }_{ m } }  })\).

When the Moon is at its perigee, Orbs = 363104 km.  We plot the GPE component graph to find GPE=0.  This graph has been zoomed to show x=3684, where GPE=0 at this value of Orbs.

From calculation, with x=3684, F_ob = 0.0000162*e^((x-3631.04)/17.37) where x is in 100 km, F_ob=0.03417 kms^-2.  From,

\(\cfrac{{v}^{2}}{{O}_{m}}={F}_{ob}\)

O_m = x + r_e = 361200+6371 = 367571 km.

v=((367571)*0.03417)^(0.5) = 112 kms^-1

Similarly,

When the Moon is at its apogee, Orbs = 406696 km. Similarly this graph has been zoomed to show x=4120 where GPE=0.

From calculation, with x=4120, F_ob = 0.0000162*e^((x- 4066.96)/17.37)  the value of Fob=0.0343 kms^-2.  From,

O_m = x + r_e =  403500+6371 = 409871 km.

v=((409871)*0.0343)^(0.5) = 118 kms^-1 

These values average to v= 115  k ms^-1, the quoted average Moon speed is 1.03  k ms^-1.

Compare to the quoted value of Moon velocity, v = 1.03 km s^-1, the calculated answer is at the next two higher order.

This is not good.  Big numbers meet with small numbers and then the exponential e....

The exponent of the exponential terms in all this expression are astronomical distances   They are large values that are difficult to measure.  The exponential being very sensitive and gives very small numbers when inverted.  These are the contributing errors to the calculations.