Depends on how fast you can push that battery. What is the delta v and at what point of the orbit are you giving it? The answer also depends, partly, on your respective masses to determine the energy given to both you and the battery.
At first, the batter will move away from you, apparently in the downward direction, After a while (about a quarter of the orbit, depending on the eccentricity of the original orbit), it will appear to turn around and come back up. It will cross slightly in front of you and go higher at about the half, orbit stage. There is a possibility of collision at the crossing point.
At about 3/4 orbit, it will start to 'fall' back down and hit you just at the point in the orbit that you pushed it away.
Reasoning: the forward velocity stays the same, but you are adding a 'vertical' component to the velocity in the downward direction. That also adds an upward vertical component to your own velocity. The new orbit for the battery is originally slightly inside of your new orbit. By Kepler's law, the inside orbit will move slightly ahead of the outer orbit while both will be elliptical (unless you can *really* push*). But that means that the battery will be slightly ahead when the orbits cross on the other side. But then, the situation is reverse, with the battery going slightly slower in its orbit and you slightly faster in yours. The two effects don't exactly cancel (depending on the energy division), but it is closer than the first crossing.
At first, the batter will move away from you, apparently in the downward direction, After a while (about a quarter of the orbit, depending on the eccentricity of the original orbit), it will appear to turn around and come back up. It will cross slightly in front of you and go higher at about the half, orbit stage. There is a possibility of collision at the crossing point.
At about 3/4 orbit, it will start to 'fall' back down and hit you just at the point in the orbit that you pushed it away.
Reasoning: the forward velocity stays the same, but you are adding a 'vertical' component to the velocity in the downward direction. That also adds an upward vertical component to your own velocity. The new orbit for the battery is originally slightly inside of your new orbit. By Kepler's law, the inside orbit will move slightly ahead of the outer orbit while both will be elliptical (unless you can *really* push*). But that means that the battery will be slightly ahead when the orbits cross on the other side. But then, the situation is reverse, with the battery going slightly slower in its orbit and you slightly faster in yours. The two effects don't exactly cancel (depending on the energy division), but it is closer than the first crossing.
