Problem 13.194

1. 250 mi A vo Fo B R = 3960 mi Problem 13.194 A shuttle is to rendezvous with a space station which is in a circular orbit at an altitude of 250 mi above the surface of the earth. The shuttle has reached an altitude of 40 mi when its engine is turned off at point B. Knowing that at that time the velocity vo of the shuttle forms an angle F = 55owith the vertical, determine the required magnitude of vo if the trajectory of the shuttle is to be tangent at A to the orbit of the space station.

2. 250 mi Problem 13.194 A vo Fo B R = 3960 mi Solving Problems on Your Own A shuttle is to rendezvous with a space station which is in a circular orbit at an altitude of 250 mi above the surface of the earth. The shuttle has reached an altitude of 40 mi when its engine is turned off at point B. Knowing that at that time the velocity vo of the shuttle forms an angle F = 55owith the vertical, determine the required magnitude of vo if the trajectory of the shuttle is to be tangent at A to the orbit of the space station. 1. Apply conservation of energy principle : When a particle moves under the action of a conservative force, the sum of the kinetic and potential energies of the particle remains constant. T1 + V1 = T2 + V2 where 1 and 2 are two positions of the particle.

3. 250 mi Problem 13.194 A vo Fo B R = 3960 mi 1 2 Solving Problems on Your Own A shuttle is to rendezvous with a space station which is in a circular orbit at an altitude of 250 mi above the surface of the earth. The shuttle has reached an altitude of 40 mi when its engine is turned off at point B. Knowing that at that time the velocity vo of the shuttle forms an angle F = 55owith the vertical, determine the required magnitude of vo if the trajectory of the shuttle is to be tangent at A to the orbit of the space station. 1a. Kinetic energy: The kinetic energy at each point on the path is given by: T = mv2

4. Problem 13.194 250 mi A vo Fo B R = 3960 mi GMm r Solving Problems on Your Own A shuttle is to rendezvous with a space station which is in a circular orbit at an altitude of 250 mi above the surface of the earth. The shuttle has reached an altitude of 40 mi when its engine is turned off at point B. Knowing that at that time the velocity vo of the shuttle forms an angle F = 55owith the vertical, determine the required magnitude of vo if the trajectory of the shuttle is to be tangent at A to the orbit of the space station. 1b. Potential energy: The potential energy of a mass located at a distance r from the center of the earth is Vg = - where G is the constant of gravitation and M the mass of the earth.

5. 250 mi Problem 13.194 A vo Fo B R = 3960 mi Solving Problems on Your Own A shuttle is to rendezvous with a space station which is in a circular orbit at an altitude of 250 mi above the surface of the earth. The shuttle has reached an altitude of 40 mi when its engine is turned off at point B. Knowing that at that time the velocity vo of the shuttle forms an angle F = 55owith the vertical, determine the required magnitude of vo if the trajectory of the shuttle is to be tangent at A to the orbit of the space station. 2. Apply conservation of angular momentum principle: For a space vehicle of mass m moving under the earth’s gravitational force: r1mv1 sinF1 = r2mv2 sin F2 where 1 and 2 represent two points along the trajectory, r is the distance to the center of the earth, v is the vehicle’s speed, and F the angle between the velocity and the radius vectors.

6. Problem 13.194 Solution 250 mi A vo Fo B GMm GMm 1 1 mvA2 - = mvo2 - rA 2 2 rB R = 3960 mi ft3 s2 (1.4077x1016 ft3/s2) (1.4077x1016 ft3/s2) 1 1 vA2 - = vo2 - (21.12x106 ft) (22.2288x106 ft) 2 2 Apply conservation of energy principle. TA + VA = TB + VB rB = ( 3960 mi + 40 mi )( 5280 ft/mi) = 21.12x106 ft rA = ( 3960 mi + 250 mi )( 5280 ft/mi) = 22.2288x106 ft GM = g R2 = (32.2 ft/s2)[(3960 mi)(5280 ft/mi)]2 = 1.4077x1016 (1)

7. Problem 13.194 Solution vA FA = 90o 250 mi A vo Fo B R = 3960 mi (21.12x106 ft) vo sin 55o = (22.2288x106 ft) vA sin 90o (2) Apply conservation of angular momentum principle. rBmvo sinFo = rAmvA sin FA Solution of equations (1) and (2) gives: vo = 12,990 ft/s