Test: Kepler’s Laws & Satellites - JEE MCQ

India's first artificial satellite was Aryabhata. Launched by soviet union on 19  April 1975

According to Kepler's Law, the square of the time period of revolution of a planet around the sun is directly proportional to the cube of semi major axis of its elliptical orbit.

T²αR³

Weight becomes zero in a freefall or when the total force on your body is zero. For example, in a freely falling lift the contact force on your body is zero, so the weighing​ machine would show zero reading. So you are weightless.

The formula for calculating the escape velocity from the surface of a celestial body (e.g. a planet) is:

where G is the universal gravitation constant, M is the planet’s mass and R is its radius. Different planets have different mass and radius - and therefore different escape velocity.

According to the second law the orbital radius and angular velocity of the planet in the elliptical orbit will vary. The planet travels faster when closer to the Sun, then slower when farther from the Sun. Hence we can say that the transverse acceleration is zero while radial and longitudinal accelerations are not zero.

As the relative velocity of the satellite with respect to the earth is zero, it appears stationary from the Earth surface and therefore it is called is geostationary satellite or geosynchronous satellite.

The Law of Periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.

The untrue statement about orbital velocity is:

1. increases with the increase in height of satellite

Explanation: Orbital velocity is the speed at which an object revolves around a planet or other celestial body in a stable orbit. According to the equation for orbital velocity, v = √(GM/r+h), where G is the gravitational constant, M is the mass of the planet, and r is the radius of the orbit.

As the height of the satellite increases (meaning it gets far to the planet), its h increase , so reasulting in decrease in velocity .

The other statements are true:

2. depends on mass and radius of planet around which it revolves: As mentioned in the equation, orbital velocity depends on both the mass (M) of the planet and the radius (r) of the orbit.

3. it is independent of mass of satellite: The mass of the satellite does not appear in the equation for orbital velocity, so it does not affect the speed at which the satellite orbits the planet.

4. decreases with an increase in radius of orbit: From the equation, we can see that as the radius of the orbit (r) increases, the orbital velocity (v) decreases.

The time period of revolution of the moon around the earth = 1 lunar month.
► T_s/T_m = (r_s/r_m)^3/2 = (1/2)^3/2
►  T_s  = (2)^-3/2 lunar month

Conservation of angular momentum:
L= 2m dA/dt