Question

The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of 5.0 earth years. Assume a 365.25-days year and MSun = 1.99 × 1030 kg.

1. What is the asteroid's orbital radius?

Express your answer in two significant figures.

2.

What is the asteroid's orbital speed?

Express answer in two significant figures.

Answer 1

1 Earth year = 365.25 days
Ms = mass of the sun = 1.99 x 10^30 kg
T = the asteroid's orbital period = 5.0 years = 157,788,000 s
G = universal gravitational constant = 6.67 x 10^-11 N∙m²/kg²
r = the asteroid's orbital radius = to be determined

(a) What is the asteroid's orbital radius?

T = 2π√(r³/GMs)
T² = [2π√(r³/GMs)]²
T² = 4π²r³/GMs
T²GMs = 4π²r³
T²GMs/4π² = r³
r³ = T²GMs/4π²
r = ³√(T²GMs/4π²)

r = ³√[(157,788,000 s)²(6.67 x 10^-11 N∙m²/kg²)(1.99 x 10^30 kg)/4π²]
r = ³√[3.304660528415952 x 10^36/4π²]
r = 437,443,907,377 m ≈ 4.37 x 10^8 km ANSWER

(b) What is the asteroid's orbital speed?

v = orbital speed of the asteroid = to be determined

v = √(GMs/r)
v = √[(6.67 x 10^-11 N∙m²/kg²)(1.99 x 10^30 kg)/(437,443,907,377 m)]
v = √(303,428,617)
v = 17,419 m/s ≈ 17.4 km/s ANSWER