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...

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

Asked on by sassy96

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valentin68 | College Teacher | (Level 3) Associate Educator

Posted on

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.

`a_("satelite") =G*M/R^2`

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

`G*M/R^2= V^2/r`

`V^2 =G*M*r/R^2`

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`

Therefore

`V=sqrt(6.67*10^-11*5.97*10^24*15/(6.7*10^6)^2)=sqrt(133)=11.54 m/s`

The speed of the plane need to be 11.54 m/s to have the same centripetal acceleration as the satellite has.

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