A 7450kg satellite has an elliptical orbit. The point on the orbit that is farthest from the earth is called the apogee. The point on the orbit that is closest to the earth is called perigee. Suppose that the speed of the satellite is 2940m/s at the apogee and 8400m/s at the perigee.
(a) Find the work done by the gravitational force when the satellite moves from the apogee to the perigee.
(b)Find the work done by the gravitational force when the satellite moves from the perigee to the apogee.
a) work done (by gravity) = potential energy difference. The potential energy at separation distance R = -G M m/R, where
M = earth mass
m = satellite mass
G = universal constant of gravity
Calculate these PE values and sbtract the apogee value from the perigee value. You should get a positive number.
The answer to (b) will be the same as (a) but with opposite sign
To solve part (a), we need to calculate the potential energy at the apogee and the perigee first.
The potential energy at any point in the satellite's orbit is given by the equation:
PE = -GMm / R
Where:
- PE is the potential energy
- G is the universal constant of gravity (approximately 6.6743 x 10^-11 N(m/kg)^2)
- M is the mass of the Earth (approximately 5.972 x 10^24 kg)
- m is the mass of the satellite (7450 kg in this case)
- R is the separation distance between the satellite and the center of the Earth.
At the apogee, the satellite is farthest from the Earth. Let's assume the separation distance at the apogee is Ra.
So, the potential energy at the apogee (PEa) is:
PEa = -GMm / Ra
At the perigee, the satellite is closest to the Earth. Let's assume the separation distance at the perigee is Rp.
So, the potential energy at the perigee (PEp) is:
PEp = -GMm / Rp
To calculate the work done by the gravitational force as the satellite moves from the apogee to the perigee, we need to find the potential energy difference:
Work done = PEp - PEa
Now, substitute the values and solve for the work done:
Work done = (-GMm / Rp) - (-GMm / Ra)
Simplify the expression:
Work done = GMm ((1/Ra) - (1/Rp))
Now you have the value for the work done when the satellite moves from the apogee to the perigee.
To solve part (b), you can use the same equation, but with the values of Ra and Rp reversed:
Work done = GMm ((1/Rp) - (1/Ra))
Remember that the work done in part (b) will have the opposite sign compared to part (a).
If you calculate the potential energy values and substitute them into the equations, you should be able to find the work done by the gravitational force for both scenarios.