>What happens when an object enters Earth's atmosphere traveling faster than its terminal velocity?

Drag slows it down toward its terminal velocity.

>As a side question, what would happen if something enters another planets atmosphere, say, Jupiter, faster than it's relative terminal velocity?

Same thing, just with its terminal velocity on Jupiter.

Terminal velocity is the speed at which gravity and atmospheric friction cancel each other out. So if an object enters the atmosphere faster then terminal velocity, earths gravity will not be able to speed up the object enough to prevent the atmosphere from slowing it down. As a result, the object would lose speed - until it either disintegrates from the heat of atmospheric friction, surface impact ('though technically that makes its speed zero) or it's speed being equal to terminal velocity.

[removed]

[removed]

To add to the other answer, terminal velocity itself depends on altitude and projectile orientation. As altitude increases, air density decreases and terminal velocity increases, towards an extremely high value in outer space much faster than any natural object. Therefore any object that has come from outer space must "enter Earth's atmosphere", something that is itself not sharply defined, faster than its terminal velocity.

Most objects hitting planetary atmospheres are not travelling straight down either. The Earth Impact Effects Program states the most common angle for meteor impacts as 45 degrees. Orbital spacecraft making a re-entry, either controlled or uncontrolled, typically enter at much shallower angles. On the other hand sounding rockets follow a trajectory that's appoximately straight up and down.

[removed]