Scientists want to place a 3600 kg satellite in orbit around Mars. They plan to

Question

Scientists want to place a 3600 kg satellite in orbit around Mars. They plan to have the satellite orbit at a speed of 2663 m/s in a perfectly circular orbit. Here is some information that may help solve this problem:

mmars = 6.4191 x 1023 kg
rmars = 3.397 x 106 m
G = 6.67428 x 10-11 N-m2/kg2

1)What radius should the satellite move at in its orbit? (Measured frrom the center of Mars.)

____________m

2)What is the force of attraction between Mars and the satellite?

___________N

3)What is the acceleration of the satellite in orbit?

__________________m/s2

5)What should the speed of the orbit be, if we want the satellite to take 8 times longer to complete one full revolution of its orbit?

___________________m/s

Explanation / Answer

Given,

m = 3600 kg ; v = 2663 m/s ; mmars = 6.4191 x 1023 kg ;
rmars = 3.397 x 106 m ; G = 6.67428 x 10-11 N-m2/kg2

1)During the circular motion the gravitational force gets balanced with the centripital force.

Fg = Fc

G M m / R2 = m V2 / R

R = G M/v2

R = 6.67 x 10-11 x 6.4191 x 1023 / 2663 x 2663

R = 6.04 x 106 m

2)The force of attraction will be:

F = G M m / R2

F = 6.67 x 10-11 x 6.4191 x 1023 x 3600 / (6.04 x 106 )2

F = 4225 N

3)The acceleration can be calculated as follows

F = m a and F = m v2/R

a = v2/R = 2663 x 2663 / 6.04 x 106 m = 1.17 m/s2

Hence, a = 1.17 m/s2

5)We know from Kepler's law

T2 = k R3

T2 = k [G M/v2]3 = k1/v6   [ k1 and k2 being the constants]

T = k2/v3 (1)

As per the conditions, T1 = 8T

T1/T = [v/v1]3 = 23

v1/v = 1/2

v1 = v/2 = 2663/2 = 1331.5 m/s

v1 = 1331.5 m/s

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