Question

An artificial satellite circles the Earth in a circular orbit at a location where the acceleration due to gravity is 9.00 m/s^2. Determine the orbital period of the satellite. I_o, a satellite of Jupiter, has an orbital period of 1.77 days and an orbital radius of 4.22 times 10^5 km. From these data, determine the mass of Jupiter. A minimum-energy transfer orbit to an outer planet consists of putting a spacecraft on an elliptical trajectory with the departure planet corresponding to the perihelion of the ellipse, or the closest point to the Sun, and the arrival planet at the aphelion, or the farthest point from the Sun. (a) Use Kepler's third law to calculate how long it would take to go from Earth to Mars on such au orbit as shown in Figure P13.19. (b) Can such an orbit be undertaken at any time? Explain.

Answer 1

g = G Me / (Re + h)^2

Re + h = r

g = G Me /r^2 = 9 m/s^2

(6.67 x 10^-11 x 5.972 x 10^24)/ r^2 = 9

r = 6.653 x 10^6 m

G Me m /r^2 = m v^2 / r

v^2 / r = G Me / r^2

v^2 = 6.653 x 10^6 x 9

v = 7737.88 m/s

T = 2 pi r / v = 5402.26 sec or 1.5 hrs

2. T = 2pi r / v

v = (2 x pi x 4.22 x 10^5 x 10^3 m) / (1.77 x 24 x 3600 s)

v = 17338.25 m/s

And G M m / r^2 = m v^2 / r

G M / r = v^2

(6.67 x 10^-11) (M) = (17338.25)^2 (4.22 x 10^8)

M = 1.902 x 10^27 kg