The answer to your question is both yes and no.
In the non-existant world where only gravity matters, yes, the
bullet would come back down at exactly the same speed as it went up.
This is because gravity works to slow things down AND to speed things
up. When the bullet is going up, gravity is trying to pull it down, so
the bullet slows down. Eventually it slows down enough that it's not
moving up any more. (This is when the bullet is at the top.) At this
point, gravity is still pulling it down, so it starts to fall. Now,
instead of slowing it down, gravity makes it go faster. Because gravity
is constant (i.e. it doesn't change), the bullet will speed up while
coming down at the SAME RATE as it slowed down when it was going up. So
when it reaches the bottom, it will be going at the same speed as it
was when it was first shot from the gun.
However, in the real world, gravity isn't the only thing effecting
the bullet. In the real world, there's also air resistance. When
gravity pulls on the bullet, it's always pulling it towards the ground,
regardless of whether that is speeding up or slowing down the bullet.
But when air restistance effects the bullet, it ALWAYS slows it down.
This makes sense, as the air is basically just pulling against the
bullet's motion. This means that in the real world, the bullet won't
come down at the same speed that it went up, because air resistance
will be slowing it down the whole way.
So, the answer is yes and no. With just gravity, yes, because the
bullet speeds up on its way down the same amount as it slowed down on
its way up. But air resistance slows it down the whole time, so in the
real world, it will be slower when it comes back down than it was when
it was first shot up.
(republished on 07/30/06)