Concept explainers
While describing a circular orbit, 185 mi above the surface of the earth, a space shuttle ejects at point A an inertial upper stage (IUS) carrying a communications satellite to be placed in a geosynchronous orbit (see Prob. 13.87) at an altitude of 22,230 mi above the surface of the earth. Determine (a) the velocity of the IUS relative to the shuttle after its engine has been fired at A, (b) the increase in velocity required at B to place the satellite in its final orbit.
Fig. P13.101
(a)
Find the velocity of the IUS relative to the shuttle after its engine has been fired at A
Answer to Problem 13.101P
The velocity of the IUS relative to the shuttle after its engine has been fired at A
Explanation of Solution
Given information:
The distance from above the earth surface to the circular orbit (x) is
The altitude of the geosynchronous orbit (A) is
The radius of the earth (R) is
The acceleration due to gravity (g) is
Calculation:
Convert the unit of radius of earth (R) from miles to feet using the relation:
Here,
Substitute
Write the expression for the force acting on the spacecraft on the surface of the earth due to gravity
Write the expression for calculating the geocentric force acting on the spacecraft when it is on the surface of earth
Here, G is the universal gravitational constant, M is the mass of the earth and
Substitute
Substitute
Write the expression for the centripetal force acting on the space shuttle carrying satellite rotating around the earth at the given altitude as follows:
Here, m is the mass of the space shuttle,
Write the expression for the geocentric force acting on the spacecraft rotating at the given altitude around the earth (F) as follows;
Equate the equations (1) and (2).
Calculate the altitude of the space shuttle from center of earth at position A
Substitute
Calculate the velocity of space shuttle at point A inertial upper stage
Substitute
Calculate the altitude of the space shuttle from center of earth at position A
Substitute
Calculate the velocity of space shuttle at point B
Substitute
Use the principle of conservation of angular momentum states that in the absence of external torque acting on the body, the angular momentum remains constant and no change of the momentum occurs during the entire process.
Find the velocity at B:
Here,
Substitute
Write the expression for the kinetic energy of the space shuttle at point A
Write the expression for the kinetic energy of the space shuttle at point B
Write the expression for the gravitational potential energy of the space shuttle at position A in the path AB
Write the expression for the gravitational potential energy of the space shuttle at position B in the path AB
Use the principle of conservation of energy states that sum of the kinetic and potential energy of a particle remains constant.
Calculate the speed of the space shuttle at position A
Write the expression for the conservation of energy as follows:
Substitute
Find the velocity at A:
Substitute
Consider the equation (1).
Find the velocity at B:
Substitute
Calculate the velocity of the IUS relative to the shuttle after the engine has been fired at point A
Substitute
Therefore, the velocity of the IUS relative to the shuttle after its engine has been fired at A
(b)
Find the increase in velocity required at B
Answer to Problem 13.101P
The increase in velocity required at B
Explanation of Solution
Given information:
The distance from above the earth surface to the circular orbit (x) is
The altitude of the geosynchronous orbit (A) is
The radius of the earth (R) is
The acceleration due to gravity (g) is
Calculation:
Calculate the increase in the velocity required at B to place the satellite in its final orbit
Substitute
Therefore, the increase in velocity required at B
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Chapter 13 Solutions
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