A satellite moves on a circular earth orbit that has a radius of 6.7x10^6 m. A model airplane is flying on a 15 m guideline on a horizontal circle.
The guideline is parallel to the ground. Find the speed of the plane such that the plane and the satellite have the same centripetal acceleration
If M is the mass of earth and m the mass of satellite the gravitational force at the satellite altitude H is
`F = G*m*M/R^2` , assuming R is measured from the center of the Earth
By writing also `F =m*g(R)` we observe that the gravitational acceleration at altitude R is
`g(R) = G*M/R^2`
which need to be equal to the satellite centripetal acceleration.
For the aircraft that is moving in a horizontal circle of radius r just above the earth, the centripetal acceleration is
`a_("plane") = V^2/r`
Both accelerations are equal means that
The numerical values are
`G =6.67*10^-11 m^3/(kg*s^2)`
`M =5.97*10^24 kg`
`R =6.7*10^6 m`
The speed of the plane need to be 11.54 m/s to have the same centripetal acceleration as the satellite has.