Askiitians Tutor Team
Last Activity: 13 Days ago
To determine the ratio of the kinetic energy required for a satellite to escape Earth's gravitational field to the kinetic energy needed for it to maintain a circular orbit just above the Earth's surface, we first need to understand the concepts of kinetic energy, gravitational potential energy, and the specific conditions for both scenarios.
Kinetic Energy for Escape Velocity
The kinetic energy (KE) required for a satellite to escape Earth's gravitational field is based on the escape velocity. The escape velocity (v_escape) from the surface of the Earth can be calculated using the formula:
v_escape = √(2GM/R)
Where:
- G is the gravitational constant, approximately 6.674 × 10-11 N(m/kg)2.
- M is the mass of the Earth, about 5.972 × 1024 kg.
- R is the radius of the Earth, roughly 6.371 × 106 m.
Substituting these values, we find:
v_escape ≈ 11,186 m/s
The kinetic energy required for escape is then given by:
KE_escape = (1/2)mv_escape2
Kinetic Energy for Circular Orbit
For a satellite in a circular orbit just above the Earth's surface, the orbital velocity (v_orbit) can be calculated using:
v_orbit = √(GM/R)
Using the same values for G, M, and R, we find:
v_orbit ≈ 7,912 m/s
The kinetic energy required for this orbit is:
KE_orbit = (1/2)mv_orbit2
Calculating the Ratio
Now, we can express the kinetic energies in terms of the mass of the satellite (m):
KE_escape = (1/2)m(11,186)2
KE_orbit = (1/2)m(7,912)2
To find the ratio of these two kinetic energies, we can simplify:
Ratio = KE_escape / KE_orbit = (11,1862) / (7,9122)
Calculating the squares:
11,1862 ≈ 125,000,000
7,9122 ≈ 62,500,000
Now, substituting these values into the ratio:
Ratio ≈ 125,000,000 / 62,500,000 = 2
Final Thoughts
The ratio of the kinetic energy required for a satellite to escape Earth's gravitational field to the kinetic energy needed to maintain a circular orbit just above the surface is approximately 2:1. This means that the energy required to escape is twice that needed for a low Earth orbit. Understanding these concepts helps in grasping the dynamics of satellite motion and the energy considerations involved in space travel.
