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If an artificial satellite is moving in a circular orbit around the earth with a speed equal to half the magnitude of the escape velocity from the earth, the height of the satellite above the surface of the earth is:
- a)2R
- b)R
- c)R/2
- d)R/4
Correct answer is option 'B'. Can you explain this answer?
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If an artificial satellite is moving in a circular orbit around the ea...
Circular Orbit and Escape Velocity
When an object is in a circular orbit around the Earth, it experiences a gravitational force that provides the necessary centripetal force to keep it in orbit. The centripetal force is given by the equation:
F_c = m * (v^2 / r)
where F_c is the centripetal force, m is the mass of the satellite, v is the velocity of the satellite, and r is the radius of the orbit.
Escape velocity is the minimum velocity an object needs to escape the gravitational pull of a celestial body. It is given by the equation:
v_e = √(2 * G * M / R)
where v_e is the escape velocity, G is the gravitational constant, M is the mass of the celestial body, and R is the radius of the celestial body.
Given Information
In this question, we are given that the speed of the satellite is equal to half the magnitude of the escape velocity from the Earth. Mathematically, this can be written as:
v = (1/2) * v_e
Analysis
To find the height of the satellite above the surface of the Earth, we need to first determine the radius of the orbit. We can rearrange the centripetal force equation to solve for r:
F_c = m * (v^2 / r)
m * (v^2 / r) = G * (m * M / r^2)
v^2 = G * (M / r)
r = G * M / v^2
Now, substitute the given value of v = (1/2) * v_e into the equation:
r = G * M / ((1/2) * v_e)^2
r = 4 * G * M / v_e^2
The height of the satellite above the surface of the Earth is then given by:
h = r - R
h = (4 * G * M / v_e^2) - R
Simplification
We can simplify the equation further by substituting the value of the escape velocity into the equation:
h = (4 * G * M / (√(2 * G * M / R))^2) - R
h = (4 * G * M / (2 * G * M / R)) - R
h = 2R - R
h = R
Conclusion
Therefore, the height of the satellite above the surface of the Earth is equal to the radius of the Earth, which is option 'B'.
When an object is in a circular orbit around the Earth, it experiences a gravitational force that provides the necessary centripetal force to keep it in orbit. The centripetal force is given by the equation:
F_c = m * (v^2 / r)
where F_c is the centripetal force, m is the mass of the satellite, v is the velocity of the satellite, and r is the radius of the orbit.
Escape velocity is the minimum velocity an object needs to escape the gravitational pull of a celestial body. It is given by the equation:
v_e = √(2 * G * M / R)
where v_e is the escape velocity, G is the gravitational constant, M is the mass of the celestial body, and R is the radius of the celestial body.
Given Information
In this question, we are given that the speed of the satellite is equal to half the magnitude of the escape velocity from the Earth. Mathematically, this can be written as:
v = (1/2) * v_e
Analysis
To find the height of the satellite above the surface of the Earth, we need to first determine the radius of the orbit. We can rearrange the centripetal force equation to solve for r:
F_c = m * (v^2 / r)
m * (v^2 / r) = G * (m * M / r^2)
v^2 = G * (M / r)
r = G * M / v^2
Now, substitute the given value of v = (1/2) * v_e into the equation:
r = G * M / ((1/2) * v_e)^2
r = 4 * G * M / v_e^2
The height of the satellite above the surface of the Earth is then given by:
h = r - R
h = (4 * G * M / v_e^2) - R
Simplification
We can simplify the equation further by substituting the value of the escape velocity into the equation:
h = (4 * G * M / (√(2 * G * M / R))^2) - R
h = (4 * G * M / (2 * G * M / R)) - R
h = 2R - R
h = R
Conclusion
Therefore, the height of the satellite above the surface of the Earth is equal to the radius of the Earth, which is option 'B'.
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If an artificial satellite is moving in a circular orbit around the earth with a speed equal to half the magnitude of the escape velocity from the earth, the height of the satellite above the surface of the earth is:a)2Rb)Rc)R/2d)R/4Correct answer is option 'B'. Can you explain this answer?
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