Reduction to One Body Problem
Let be the mass of the satellite, be the mass of Jupiter, be the vector from the satellite to the planet, and be the angle between the satellite and Jupiter where We will proceed to use polar coordinates in our calculations. The kinetic energy and potential energy of the system are and respectively. Therefore, the Lagrangian is . Applying the Euler-Lagrange equation to the coordinate, we have
Applying the Euler-Lagrange equation on the coordinate, we have
From equation , we have that . Note that since angular momentum is conserved in this system, . Substituting this into equation , we get . Integrating with respect to time, we have
which is a statement of conservation of energy. Solving in terms of , we have
Note that this is equivalent to
Rewriting the conservation of angular momentum, we have Substituting this into equation , we have
Integrating both sides, we have
where is the initial angle. Evaluating this integral, we have
We can rewrite this as
Recognizing this as the general equation of a conic, we know that the eccentricity of the orbit is determined by
Solution of 2001 AP ® PHYSICS C: MECHANICS FREE-RESPONSE QUESTION Mech 2. Part C
(i) If the speed of the satellite is slightly faster, note that the eccentricity will be , implying an elliptical orbit. Additionally, note that . If the satellite is slightly faster, will increase and thus will increase except for when , where , that is, when the satellite passes the injection point.
(ii) If the speed of the satellite is slightly slower, note that the eccentricity will be , implying an elliptical orbit. Additionally, note that . If the satellite is slightly slower, will decrease and thus will decrease except for when , where , that is, when the satellite passes the injection point.