Gravitation and Orbital Mechanics

According to Kepler's Second Law, a planet sweeps out equal areas in equal times, which requires it to move faster when it is closer to the Sun (perihelion) and slower when it is farther away (aphelion).

Kepler's Third Law states that the square of the orbital period (T^2) is proportional to the cube of the semi-major axis (r^3). Therefore, T is proportional to the square root of r^3.

Venus has an orbital period of approximately 224.7 Earth days, which is commonly rounded to 225 days in educational contexts.

Gravity follows the inverse-square law, meaning force is proportional to 1/r^2. If the distance is halved, the force increases by a factor of 2^2, which is 4.

In the formula for orbital velocity, m represents the mass of the central body being orbited. Since Venus orbits the Sun, m is the mass of the Sun.

One astronomical unit (AU) is defined as the average distance between the Earth and the Sun, approximately 150 million kilometers.

An orbiting planet maintains its path due to the balance between gravitational pull and its orbital velocity. If a collision significantly increases the planet's orbital velocity, it can exceed the escape velocity required to maintain a closed orbit, causing it to leave the system.

The Earth orbits the Sun at an average speed of approximately 18.5 miles per second (about 30 kilometers per second).

To maintain a stable low Earth orbit, the Space Shuttle had to travel at approximately 17,500 mph to balance gravitational pull with centrifugal force.

A satellite in orbit is in a state of continuous free fall toward the Earth. Because the body inside the satellite is also in free fall, it experiences weightlessness regardless of the orbital radius. Therefore, the reading of the spring balance will be zero in both cases, making W1 equal to W2.

Given :  $R = 7\times 10^8 \ m$     and    $M = 2\times 10^{30} \ kg$
Volume of sun  $V = \dfrac{4}{3}\pi R^3 = \dfrac{4}{3}\pi\times (7\times 10^8)^3 = 1.4\times 10^{27} \ m^3$
Density of sun   $\rho = \dfrac{M}{V} = \dfrac{2\times 10^{30}}{1.4\times 10^{27}} = 1.43\times 10^3 \ kg/m^3$
Thus order of magnitude of density of sun is $10^3 \ kg/m^3$.

This question reflects the concept of gravitational time dilation where time for the crew is experienced at a slower rate than for those outside the gravitational field.

A satellite is any object that orbits another, larger object. This can be natural (like the moon) or artificial.

Earth completes one full orbit around the sun in approximately 365 days. This period defines our calendar year.

An orbit is the curved path that an object in space takes around another object due to gravity. An axis is a line of rotation, while rotation is the act of spinning.