A planet in another solar system orbits a star with a mass of 4.0 x 10³⁰ kg.At one point in its orbit it is 250 x 10⁶ km from the star and is moving at 35 km/s.Take the universal gravitational constant to be 6.67 x 10^-11 m²/s² .kg and calculate the semimajor axis of the planet's orbit.The result is: A)79 x 10⁶ km B)140 x 10⁶ km C)290 x 10⁶ km D)320 x 10⁶ km E)590 x10⁶ km

Question

Quizplus · Accepted Answer

We can use Kepler's third law to solve for the semimajor axis:

T^2 = (4π^2/ GM) a^3

where T is the period of the planet's orbit, G is the gravitational constant, M is the mass of the star, and a is the semimajor axis.

We can first solve for the period of the orbit:
r = 250 x 10^6 m = 250 x 10^3 km

v = 35 km/s

From conservation of energy we can write:

(1/2)mv^2 - GMm/r = (1/2)mv2' - GMm/r'

where r' is the distance from the star at another point in the orbit, and v' is the velocity at that point.

We can set r' equal to the semimajor axis a, and v'=0 at the furthest point in the orbit:

(1/2)(35 km/s)^2 - (6.67x10^-11 N(m^2/kg^2))(4x10^30 kg)/250x10^9 m = -(6.67x10^-11 N(m^2/kg^2))(4x10^30 kg)/a

Solving for a, we get:

a = 290 x 10^6 m = 290 x 10^3 km

Therefore, the answer is C) 290 x 10^3 km.

A Planet in Another Solar System Orbits a Star with a Mass