Sputnik I was launched into orbit around Earth in 1957. It had a perigee (the closest approach to Earth, measured from Earth’s center) of 6.81 x...

Hello!

When a relatively light body rotates around much heavier body, its orbit is elliptical with the heavier object at one focus of this ellipse. And the speed of a smaller object is

`v=sqrt(G*M*(2/r-1/a)),`

where `G` is the constant of universe gravitation, `M` is the mass of a heavier body, `r` is the current distance between objects and `a` is the major semi-axis of the elliptic orbit.

Denote the minimal distance as `r_p` and the maximum as `r_a.` Then `r_p+r_a=2a,` or

`a = (r_p+r_a)/2.`

Also, at the perigee `r=r_p.` And we know `G` and `M.`

The final formula is

`v_p=sqrt(G*M*(2/r_p-2/(r_p+r_a))).`

In numbers,

`v_p=sqrt(5.97*10^24*6.67*10^(-11)*2*(1/(6.81*10^6)+1/(6.81*10^6+7.53*10^6))),`

which is approximately

`sqrt(79.64*10^7*(0.147+0.070)) approx 13141(m/s).`

P.S. Sources give the completely different values for `r_p` and `r_a.`

P.P.S. The main formula may be derived from the basic physical laws, see the first link.