Question

1. how does the earths speed change during different parts of the orbit? which of keplers laws describes this change?
2. pause the simulation. click the “clear” button at the bottom to reset the “earth days”. play the simulation and record approximately how many days it takes the earth to complete one full orbit around the sun with a 1.5 mass.
3. how did the distance and orbital period change when the suns mass was increased?

Explanation:

10.

Step1: Recall Kepler's laws

Kepler's second law (law of equal - areas) states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

Step2: Analyze Earth's speed

When the Earth is closer to the Sun (perihelion), it moves faster because to sweep out the same - area in the same time interval as when it is farther from the Sun (aphelion), it must cover a greater arc length. When it is farther from the Sun (aphelion), it moves more slowly.

Step1: Recall Kepler's third law

Kepler's third law is $T^{2}=\frac{4\pi^{2}}{GM}a^{3}$, where $T$ is the orbital period, $G$ is the gravitational constant, $M$ is the mass of the Sun, and $a$ is the semi - major axis of the orbit.

Step2: Analyze the effect of increasing the Sun's mass

As the mass $M$ of the Sun increases, from the formula $T^{2}=\frac{4\pi^{2}}{GM}a^{3}$, the orbital period $T$ decreases. Also, according to Newtonian gravity, the gravitational force $F = \frac{GMm}{r^{2}}$ between the Sun and the Earth. A more massive Sun would pull the Earth in more strongly, causing the semi - major axis $a$ (average distance) of the Earth's orbit to decrease.

Answer:

The Earth moves faster when it is closer to the Sun (perihelion) and slower when it is farther from the Sun (aphelion). Kepler's second law (law of equal - areas) describes this change.

11.

The actual time it takes for the Earth to complete one full orbit around the Sun is approximately 365.24 days. Changing the mass of the Sun to 1.5 times its actual mass would require running the simulation as described in the problem. However, according to Kepler's third law ($T^{2}\propto a^{3}$ where $T$ is the orbital period and $a$ is the semi - major axis), a more massive Sun would cause the Earth's orbital period to decrease. But without running the simulation, we can't give the exact value from the simulation. If we assume a theoretical calculation based on Kepler's third law and Newtonian gravity, we would need to know more details about the initial conditions of the simulation. But if we consider the real - world Earth - Sun system as a reference for the simulation, we know that for a 1.5 mass Sun, the period would be less than 365.24 days.

12.