Potential Energy on surface of earth = -

Total energy = mv₀² -

Kinetic Energy at a height h = mv²

Potential Energy at this height = -

Total energy = mv² -

By the principle of conservation of energy
mv² - = mv₀² -

(v₀² - v²) = -

But GM = gR²

(v₀² - v²) =

v₀² - v² =

For maximum height v = 0

v₀ = 90% of escape speed = 0.9


0.81R + 0.81h_max = h_max
0.19h_max = 0.81R
h_max =

= 4.26R

3. Two satellites A and B of equal masses, move in the equitorial plane of earth, close to the earth's surface. Satellite A moves in the same direction as that of the rotation of earth while satellite B moves in the opposite direction. Determine the ratio of the kinetic energy of B to that of A in the reference frame fixed to earth (g = 9.8 m/s²)

Solution:

Velocity of A with respect to earth =

- wR
Velocity of B with respect to earth =

+ wR

= 1.265
where T = 24 x 60 x 60 s     R = 6400 x 10³m
HARD
:

1. An artificial satellite of mass m of a planet of mass M, resolves in a circular orbit whose radius is n times the radius R of the planet. In the process of motion, the satellite experiences a slight resistance due to cosmic dust. Assuming resistance force on satellite depends on velocity as F = av²
where a is constant. Calculate how long the satellite will stay in orbit before it falls onto the planet's surface.
Solution: Total energy in circular orbit of radius r,
E = -

Rate of change of energy =

= F_resistiv. v = av².v = av³

also gravity provides the centripetal force


Now





2. Find the maximum and minimum distances of the planet A from the sun S if at a certain moment of time it was a distance r₀ and travelling with the velocity vector being equal to

.
Solution:

At minimum and maximum distance velocity of satellite makes on angle of 90⁰
with radius vector.
Applying conservation of angular momentum
mv₀r₀sin= mvr

By energy conservation


Solving equation (i) and (ii) we get two values of r, one is maximum distance another is minimum distance.

r_max=


r_min=


where


Key words
:

• Gravitational Force.
• Gravitationl Field.
• Gravitational Potential.
• Escape Speed.
• Orbittal Speed.
• Areal Velocity.
• Parking Orbit.