Question 1

According to Kepler's First Law, what is the shape of a planet's orbit around the Sun, and where is the Sun located within this shape?

Accepted Answer

An elliptical orbit with the Sun at one of the foci

Question 2

Kepler's Second Law states that a planet sweeps out equal areas in equal times. What does this imply about the speed of a planet in its orbit?

Accepted Answer

The planet moves faster when it is closer to the Sun

Question 3

Given Kepler's Third Law for objects orbiting the Sun, $$\frac{T^2$}{R^3$} = 1$$, where $$T is $$the orbital period in years and $$R is $$the average radius in astronomical units (au), what is the orbital period of a planet with an average orbital radius of $$4 au$$?

Accepted Answer

$$T = 8$$ years (since $$T^2$$ = $$R^3$$, $$T = \sqrt{4^3$} = \sqrt{64} = 8)$$

Question 4

Which planet in our Solar System has the fastest average orbital speed, and why?

Accepted Answer

Mercury, because it is closest to the Sun and must travel faster to balance the Sun's gravity

Question 5

If a newly discovered planet orbits the Sun at an average distance of $$9 au$$, what is its orbital period according to Kepler's Third Law?

Accepted Answer

$$T = 27$$^{1/2}$$ = 5.2$$ years (since $$T^2$$ = $$R^3$$, $$T = \sqrt{9^3$} = \sqrt{729} = 27)$$

Question 6

A comet travels in a highly elliptical orbit around the Sun. According to Kepler's Second Law, how does the comet's speed change as it moves from aphelion (farthest point) to perihelion (closest point)?

Accepted Answer

The comet's speed increases as it approaches perihelion

Question 7

If the average distance between a planet and the Sun doubles, by what factor does its orbital period change according to Kepler's Third Law?

Accepted Answer

It increases by a factor of $$2^{3/2} = 2.83$

Question 8

Which of the following best explains why the ratio $$\frac{T^2$}{R^3$} is $$approximately $$1$$ for all planets in our Solar System?

Accepted Answer

All planets orbit the Sun, so the constant is the same for each

Question 9

A satellite orbits the Sun at $$0.25 au. $$What is its orbital period in years?

Accepted Answer

$$T = \sqrt{(0.25)^3$} = \sqrt{0.015625} = 0.125$$ years

Question 10

If Pluto's average distance from the Sun is $$39.44 au$$, what is its orbital period according to Kepler's Third Law?

Accepted Answer

$$T = 248$$ years

Question 11

Which of the following statements is a direct consequence of Kepler's First Law?

Accepted Answer

The distance between a planet and the Sun changes as the planet moves along its orbit

Question 12

A planet has an orbital period of $$8$$ years. What is its average distance from the Sun in astronomical units?

Accepted Answer

$$R = (8^2$)^{1/3}$$ = $$(64)^{1/3} = 4$ au

Question 13

Which of the following best describes the relationship between a planet's orbital period and its average distance from the Sun according to Kepler's Third Law?

Accepted Answer

The square of the orbital period is proportional to the cube of the average distance

Question 14

If a planet's orbit is highly eccentric, which of Kepler's Laws explains why the planet spends more time far from the Sun than close to it?

Accepted Answer

Kepler's Second Law

Question 15

A spacecraft is sent to orbit the Sun at $$2 au. $$How long will it take to complete one orbit?

Accepted Answer

$$T = 2.83$$ years$$\ (T = \sqrt{2^3$} = \sqrt{8} = 2.83)$$

Question 16

Which of the following is NOT a correct statement about Kepler's Laws?

Accepted Answer

All planets move at the same speed in their orbits

Question 17

If the average distance from the Sun to Saturn is $$9.54 au$$, what is Saturn's orbital period?

Accepted Answer

$$T = 29.5$$ years

Question 18

A planet orbits a star other than the Sun. Its $$\frac{T^2$}{R^3$}$$ ratio is found to be $$2. $$What does this indicate?

Accepted Answer

The planet is not orbiting the Sun, so the constant is different

Question 19

Which of the following real-world observations provided evidence for Kepler's First Law?

Accepted Answer

The varying distance between Earth and the Sun throughout the year

Question 20

If a planet's orbital period is $$27$$ years, what is its average orbital radius in au?

Accepted Answer

$$R = (27^2$)^{1/3}$$ = $$(729)^{1/3} = 9$ au

Kepler's Laws and Planetary Motion

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