When a Satellite Is near Earth, Its Orbital Trajectory May $V$

Multiple Choice

When a satellite is near Earth, its orbital trajectory may trace out a hyperbola, a parabola, or an ellipse. The type of trajectory depends on the satellite's velocity $V$ in meters per second. It will be hyperbolic if $V > \frac { k } { \sqrt { D } }$ , parabolic if $\mathrm { V } = \frac { \mathrm { k } } { \sqrt { \mathrm { D } } }$ , and elliptical if $\mathrm { V } < \frac { \mathrm { k } } { \sqrt { \mathrm { D } } }$ , where $\mathrm { k } = 2.82 \times 10 ^ { 7 }$ is a constant and $\mathrm { D }$ is the distance in meters from the satellite to the center of Earth. Solve the problem.
-If a satellite is scheduled to leave Earth's gravitational influence, its velocity must be increased so that its trajectory changes from elliptical to hyperbolic. Determine the minimum increase in velocity necessary for a
Satellite traveling at a velocity of 5696 meters per second to escape Earth's gravitational influence when $\mathrm { D } = 167 \times 10 ^ { 6 } \mathrm {~m} .$

A) $- 3514 \mathrm {~m}$ per sec
B) $5765 \mathrm {~m}$ per sec
C) $7878 \mathrm {~m}$ per sec
D) $5627 \mathrm {~m}$ per sec