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Old Dec 9th 2010, 02:02 PM   #1
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Newton and motion of satellites

I need help with the second part of a two part question.

Using Newton's thought on the motion of satellites:

a. Calculate the speed that a satellite shot from the cannon must have in order to orbit Earth 150 km above its surface. The mass of the Earth is 5.97 x 10²⁴ kg and the radius of the Earth is 6.52 x 10⁶ m.

I got this:

V=((6.7 x 10-ıı)(5.97 x 10²⁴)/(6.52 x 10⁶))
V=7,832.50 m/s

Is that correct?

b. How long, in minutes, would it take for the satellite to complete one orbit and return to the cannon?

I have so far:

T=2pi((6.52 x 10⁶)³/(6.7 x 10-ıı)(5.97 x 10²⁴))
T=5235
Is 5235 in m/s? If so how do i get it into minutes?
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Old Dec 10th 2010, 11:25 PM   #2
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a)(1/2)mv^2=GMm/r^2


substitute the values

Last edited by BAdhi; Dec 10th 2010 at 11:28 PM.
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Old Dec 11th 2010, 12:48 AM   #3
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a. I do not agree. The formula for the gravitational potential energy is

U = -GMm/r

The change in gravitational energy equals the initial kinetic energy provided.

Hence, this gives

(mv^2)/2 = - (GMm)/(r+h) + (GMm)/r

b. The centripetal force on the satellite is provided by the gravitational force.

Hence;

mRω^2 = (GMm)/(R^2)

Simplifying gives indeed

T = 2pi \sqrt{R^3/(GM)}

And here, R = r + h

And the T that you got is in seconds, not m/s

And one assumption here is that the Earth is not rotating on itself then. Because otherwise, you'll have another value, lower or higher depending whether it is rotating in the same direction as the Earth or rotating in the opposite direction.
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