Suppose there's some satellite orbiting the earth in circular motion. Suppose there's an asteroid that hits the satellite in the same direction as the instant velocity vector of the satellite. The collision causes the satellite to move faster. And here are my 2 questions:
1) Why will the satellite start moving in an elliptical orbit? Is it because its speed has increased, but the centripetal acceleration hasn't? It seems intuitively right for me, because then the satellite covers larger path before the centripetal acceleration causes the change in the velocity vector direction.
2) However, what confuses me is the fact that the centripetal acceleration is dependent on the velocity ($v^2/r$). On the other hand, since the force of gravity exerted on the satellite by the earth, hasn't changed after the hit (if we neglect asteroid's mass or suppose it just fell down after the hit), because there was no change in the radius (i suppose) of the circular path, there was no change in the centripetal acceleration. So these two arguments contradict each other. Where am I wrong? What I miss here?
Thanks in advance.
Best Answer
If an object is moving in a circular motion, its velocity $\vec{v}$ changes. The centripetal acceleration is just a formula that gives you the length of the derivative $\frac{d\vec{v}}{dt}$ which is the acceleration. It must be caused by some force, according to Newton's second law. If you are holding the object with a rope, then it is the tension of the rope, if it is a satelite on a circular orbit, then the force is of gravitational nature.
When the asteroid hits the satellite, $\vec{v}$ changes, while the gravitional force remains the same. So, the force now creates the same acceleration, but now it does not coincide with 'centripetal acceleration' for this speed (which is just a number characterizing the orbit, not the object). This simply means that the object will leave the circular orbit, because its acceleration and speed now correspond to a different trajectory. This trajectory happens to be elliptic/parabolic/hyperbolic depending on the speed. These cases can be distinguished by total energy -- $E<0$, $E=0$, $E>0$ respectively.