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Question 4070: Please help I am not sure how to solve the following!
s=1/2 gt^2 (solve for g)
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! Let the altitude be h and the radius of the earch be R.
Also, assume the velociety the of the satellite is V
the central force mV^2/(R+h) = GmM/(R+h)^2
or V^2 = GM/(R+h)
so V = sqrt(GM/(R+h))...(1)
(where M is the mass of the earth , m is the mass of the satellite
G is the gravitational constant))
The length of the orbit of the satellite is
2pi(R+h) = V* 1.5 ( 90 min = 1.5 hrs)
Solve 2pi(R+h) = 1.5 sqrt(GM/(R+h))...(2)
for h (by taking squares on both sides) then you can get the answer
of h(altitude)
Let T be the time for a satellite to orbit (of radius R+h) the earth.
Use (1), we have
T = 2pi (R+h)/ V = 2pi (R+h) /sqrt(GM/(R+h)) = 2pi(R+h)^(3/2)/sqrt(GM)
so T is an increasing function of h.
When h = 0, T has min. value 2pi(R)^(3/2)/sqrt(GM).
But since mg = mGM/R^2 = m V^2/R. (g =9.8 meter/sec^2)
The corresponding V = sqrt(Rg).
So, T has min value = 2pi R/ V = 2pi R/ sqrt(Rg) = 2pi sqrt(R/g)
Of course, you have to find the values of R,M & G by yourself to solve
(2) and find min T.
Kenny
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