A 7300-kg satellite has an elliptical orbit, as in the figure. The point on the orbit that is farthest from the earth is called the apogee and is at the far right side of the drawing. The point on the orbit that is closest to the earth is called the perigee and is at the far left side of the drawing. Suppose that the speed of the satellite is 2730 m/s at the apogee and 8450 m/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.
To find the work done by the gravitational force as the satellite moves from the apogee to the perigee and vice versa, we need to calculate the change in gravitational potential energy.
The gravitational potential energy (U) is given by the formula:
U = - G * (m * M) / r
where G is the gravitational constant (6.67430 x 10^-11 N m^2/kg^2), m is the mass of the satellite (7300 kg), M is the mass of the Earth (5.972 x 10^24 kg), and r is the distance between the satellite and the center of the Earth.
Considering the apogee as point A and the perigee as point B, we will calculate the work done in two parts:
(a) From A (apogee) to B (perigee):
The change in gravitational potential energy is given by:
ΔU = Uperigee - Uapogee
Substituting the values, we get:
ΔU = (- G * (m * M) / rperigee) - (- G * (m * M) / rapogee)
Now, we need to find the distances rperigee and rapogee. In an elliptical orbit, the distance is not constant, so we need to find the radius at each point.
The speed of the satellite (v) is given as 2730 m/s at the apogee and 8450 m/s at the perigee. Using the formula for the speed of a satellite in an elliptical orbit:
v = √(G * M * ((2/r) - (1/a)))
where a is the semi-major axis of the ellipse (the average of the distances from the center of the ellipse to the apogee and perigee). Rearranging the formula, we have:
r = 2 / ((v^2 * M) / (G * M) + 1/a)
Substituting the values, we can find rperigee and rapogee.
Once we have the values of rperigee and rapogee, we can substitute them into the equation for ΔU to find the change in gravitational potential energy.
(b) From B (perigee) to A (apogee):
Using the same approach as in part (a), we can calculate the change in gravitational potential energy using the formula:
ΔU = Uapogee - Uperigee
Substituting the values of rapogee and rperigee, we can find the change in gravitational potential energy.
Remember, work done is the negative change in potential energy, so to find the work done by the gravitational force, simply multiply the change in potential energy by -1.