A planet orbits a star with the path shown below.

Question

A planet orbits a star with the path shown below.

image 1

a. On the axis below, draw the general shape of the graph of the planet's gravitational potential energy as it completes one orbit. Start the graph where the planet is shown in the diagram above. (2 points)

image 2(GPE on y axis and t on x axis)

b. In general, the gravitational potential energy for a planet orbiting a star is given by U=-Gm(planet)m(star)/r . The maximum distance of this planet from its star is 700,000,000 km, and the minimum distance is 600,000,000 km. The mass of the planet is 4 × 10²² kg, and the mass of the star is 5 × 10³¹ kg. How much does the GPE change during one full orbit? (3 points)

c. A comet of mass 2 × 10⁸ kg is pulled toward the star. If the comet's inital velocity is very small, and the comet starts moving toward the star from 700,000,000 km away, how fast is it going right before it hits the surface of the star? (Assume that it does not lose any mass by melting as it approaches the star.) (3 points)

Answer 1

Here are the approaches for parts (b) and (c).

(b)

U = GPE

U = -GMm / r

Convert the 2 extreme (max and min) values of r to meters and plug them into the equation for U. The 2 resulting values of U define the range over which the GPE varies along the orbit. But keep in mind that after each full orbit, the GPE returns to its initial value.

(c)

To answer this question, we need to know the radius of the star, which was not given in the problem. Work with energy conservation. Assume the comet of mass m started from rest at a distance x from the star. Assume the radius of the star is R.

mv² / 2 - GMm/R = m(0)² / 2 - GMm / x

Note that m cancels out and you are left with

v² / 2 - GM / R = -GM / x ⇒

v = √[2GM(1/R - 1/x)]

You know M in kg. You know x in km. Convert x to meters. You need R in meters, which is not stated in the problem. Then plug the numbers into the equation for v.