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...

During its movement, the satellite is subjected to a centrifugal force, pointing towards the outside of the trayectory. In addition, is subjected to the force of Earth's gravity.

To keep the satellite in orbit, the speed must be such that, the centrifugal force it is equal and opposite to the force of gravity; so we can consider the following equality:

Fc = Fg

(m*v^2)/r = (G*M*m)/r^2

Where:

M,  is the mass of the earth.

m,  is the mass of the satellite.

r,  is the distance between the satellite and the earth

G,  is the gravitational constant.

v,  is the tangential velocity of satellite.

Solving, for the speed at perigee:

v^2 = (G*M)/r

v = sqrt (GM/r) = sqrt [(6.67*10-11)(5.97*10^24)/(6.81*10^6)]

v = 7.64*10^3 m/s  =  7.64 km/s

 So, at perigee the satellite must have a speed of 7.64 km/s