In an accidental explosion, a satellite breaks in half while while in a circular orbit about earth. One half is brought to momentarily to rest. What is the fate of the half brought to rest? What happens to the other half?

The piece brought to rest falls straight down to Earth (and probably burns up along the way).

The ither part doubles its velocity in the same direction I(thus maintaining total momentum), making the orbit an ellipse with apogee much farther from Earth; and perigee where the exoplosion occurred.

When a satellite breaks in half, the fate of each half will depend on the specific dynamics of the situation. In this case, one half is brought to a momentary rest while the other continues to orbit the Earth.

The half that is brought to rest in space will no longer be under the influence of Earth's gravitational force. Without the continuous pull of gravity, the satellite half will slowly drift in space along the path dictated by its initial momentum. It will essentially continue moving in a straight line at a constant velocity unless acted upon by other forces, such as the gravitational pull of another celestial body or collisions with other objects in space.

On the other hand, the other half of the satellite that continues to orbit Earth will follow its original circular path, albeit with a slightly altered orbit. Since the two halves have different masses and velocities, their individual orbits will be affected differently. The half that remains in orbit will experience a change in its center of mass, causing it to deviate from its original circular orbit slightly.

Over time, the gravitational forces exerted by the Earth and other celestial bodies, as well as atmospheric drag, may further alter the orbit of the remaining half. It could eventually decay and reenter the Earth's atmosphere, burning up as it encounters the friction from the air molecules during reentry. Alternatively, if the satellite remains above the Earth's atmosphere, it may continue to orbit for an extended period or collide with other space debris or satellites.

The fate of the two halves of the satellite ultimately depends on multiple factors such as their initial conditions, location, and interactions with other objects in space.