Question

Can someone help me with this question and explain to me how to solve this?

Planet A takes one year to go around the sun at a distance of one A U. (astronomical unit). Planet B is three A.U. from the sun. How many years does Planet B take to orbit? Choose the closest answer.
3 years
5 years
7 years
9 years

Answer 1

5 years

Step-by-step explanation:

The centripetal acceleration of a planet is equal to the acceleration due to gravity.

ac = g

Centripetal acceleration is:

ac = v² / r

where v is velocity and r is radius of travel.

Acceleration due to gravity is:

g = GM / r²

where G is gravitational constant, M is the mass of the sun, and r is the radius of travel.

Therefore:

v² / r = GM / r²

v² = GM / r

v = √(GM / r)

Distance is speed times time, so:

d = v t

2πr = √(GM / r) t

t = 2πr √(r / (GM))

t = 2π √(r³) / √(GM)

We know that when r = 1 AU, t = 1 year.

1 = 2π √(1³) / √(GM)

1 = 2π / √(GM)

2π = √(GM)

Substituting:

t = 2π √(r³) / (2π)

t = √(r³)

When r = 3AU:

t = √(3³)

t = 5.2

Planet B takes approximately 5 years to orbit the sun.