PS 05.5 Kepler's Laws of Planetary Motion

What shape is the path of each planet around the sun according to Kepler's First Law?

• Circle

• Ellipse

• Parabola

• Hyperbola

Answer explanation

According to Kepler's First Law, the path of each planet around the sun is an ellipse, with the sun at one of the foci. This means that the correct answer is 'Ellipse', not a circle, parabola, or hyperbola.

According to Kepler's Second Law, what does a line from the sun to a planet sweep out in equal times?

• Equal distances

• Equal areas

• Equal angles

• Equal velocities

Answer explanation

According to Kepler's Second Law, a line from the sun to a planet sweeps out equal areas in equal times. This means that the area covered by the line in a given time interval is constant, regardless of the planet's distance from the sun.

Kepler's Third Law involves the ratio of the squares of the periods of any two planets. What is this ratio equal to?

• The ratio of the cubes of their average distances from the sun

• The ratio of their masses

• The ratio of their velocities

• The ratio of their diameters

Answer explanation

Kepler's Third Law states that the square of the orbital period of a planet is proportional to the cube of its average distance from the sun. Thus, the correct choice is the ratio of the cubes of their average distances from the sun.

Explain why the sun appears to be at the center of Earth's orbit despite Kepler's First Law stating it is at one focus.

• Earth's orbit is a perfect circle

• Earth's orbit is nearly circular, making the foci appear as one

• The sun's gravitational pull is stronger at the center

• The sun's light distorts our perception

Answer explanation

Earth's orbit is nearly circular, which makes the two foci of the ellipse (one being the sun) appear very close together. This visual effect leads us to perceive the sun as being at the center of Earth's orbit.

If a planet's orbit is highly elliptical, how would Kepler's Second Law affect the speed of the planet at different points in its orbit?

• The planet moves at a constant speed

• The planet moves faster when closer to the sun and slower when farther

• The planet moves slower when closer to the sun and faster when farther

• The planet's speed is unaffected by its distance from the sun

Answer explanation

According to Kepler's Second Law, a planet sweeps out equal areas in equal times. This means it moves faster when closer to the sun (perihelion) and slower when farther away (aphelion), confirming the correct choice.

Answer explanation

According to Kepler's Third Law, the square of the period of a planet is proportional to the cube of its average distance from the sun. If one planet is twice as far, then \(T_2^2 = (2^3)T_1^2 = 8T_1^2\), leading to \(T_2 = 4T_1\).

Describe how Kepler's Second Law can be used to determine the relative speed of a planet at different points in its orbit.

• By measuring the distance traveled in a fixed time

• By comparing the areas swept out in equal times

• By calculating the angle of the orbit

• By observing the planet's brightness

Answer explanation

Kepler's Second Law states that a line segment joining a planet and the Sun sweeps out equal areas in equal times. By comparing the areas swept out in equal time intervals, we can determine the planet's relative speed at different points in its orbit.

• 1

• 2

• 0.5

• 4

Answer explanation

Using Kepler's Third Law, we calculate \( \frac{T^2}{R^3} = \frac{8^2}{4^3} = \frac{64}{64} = 1 \). Thus, the ratio is 1, confirming the law. The correct answer is 1.

How does Kepler's Third Law help in understanding the motion of moons around planets?

• It shows that moons have the same period as their planets

• It provides a relationship between the period and distance of moons from their planets

• It explains why moons have elliptical orbits

• It predicts the gravitational pull of the planet on its moons

Answer explanation

Kepler's Third Law establishes a relationship between the orbital period of moons and their distance from the planet. This helps us understand how the distance affects the time it takes for a moon to complete its orbit.

A planet orbits the sun with a period of 27 years. If another planet orbits at an average distance 3 times greater, what is its orbital period according to Kepler's Third Law?

• 81 years

• 9 years

• 243 years

• 27 years

Answer explanation

According to Kepler's Third Law, the square of the orbital period (T) is proportional to the cube of the average distance (R) from the sun. If R is 3 times greater, T^2 = 3^3 = 27, so T = √27 = 3√3, which is 81 years.