Question

1. Which of the following does Kepler's Second Law support?

Answer 1

Answer:

A: When a planet is closer to the Sun, its speed is greater than when it is farther away.

Explanation:

Kepler's laws establish that:

• All the planets revolve around the Sun in an elipse orbit, with the sun in one of the focus (Kepler's first law).

• A planet describes equal areas in equal times (Kepler's second law).

• The square of the period of a planet will be proportional to the cube of the semi-major axis of its orbit (Kepler's third law).

          $T^{2} = a^{3}$

       Where T is the period of revolution and a is the semi-major axis.

Since planets orbit around the Sun in an ellipse, with the Sun in one of the focus, in some moments of their orbit they will be closer to the Sun (known as perihelion). According with Kepler's second law to complete the same area in the same time, they has to speed up at their perihelion and slow down at their aphelion (point farther from the sun in their orbit).

That increase in the orbital velocity as a consequence of the distance can be prove by the Universal law of gravitation:

$F = G\frac{Mm}{r^{2}}$   (1)

Where G is the gravitational constant, M and m are the masses of the two objects and r is the distance between them.

$F = ma$   (2)

Where m is the mass and a is the acceleration. Equation (2) can be replaced in equation (1).

$ma = G\frac{Mm}{r^{2}}$    (3)

Since it is a circular motion, the centripetal acceleration is defined as:

$a_{c} = \frac{v^{2}}{r}$   (4)

Replacing (4) in (3) it is got:

$m\frac{v^{2}}{r} = G\frac{Mm}{r^{2}}$   (5)

Notice that m in equation (5) represents the mass of the planet while M is the mass of the Sun.

Equation (5) can be expressed in term of v:

$v = \sqrt{\frac{GM}{r}}$   (6)

Hence, if the distance increases the orbital velocity decreases (inversely proportional).