Their final Velocity becomes their orbital Velocity and that leaves it at the corresponding altitude

The rocket motor burns during the entire launch, this allows the orbit to be continually trimmed from it's original elliptical trajectory at liftoff, to it's desired circular orbit at the time the second stage cuts out.

Cheers!

" why do rockets during launch automatically place spacecraft into a circular orbit?" They don't, the desired orbit is carefully calculated, and the launch rockets are specifically designed to achieve the target orbit, there is nothing "automatic" about it.

Without any course corrections, most *are* launched into elliptical orbits. Apollo, for example, would fire engines for about 12 1/2 minutes, and at the end of their boost, they’d be in an elliptical orbit from about 100 to 130 miles high. Apollo 9, which stayed in Earth orbit, fired engines at apogee to make their orbit more circular.

It can be done with a single engine firing - but it requires the spacecraft to maneuver as it’s being boosted, such that the trajectory is already circular at the end of it’s boost phase.

Could be overlooking that getting a rocket up is one part, trimming it for relative earth orbit comes after that.

Proper trajectories and burn timing, many rockets need little in the way of ' orbital ops' for orbital insertion.

Ever hear the term launch window, that plays a roll.

Consider, orbit is like freefall. A rockets speed sees that fall holds an orbit and misses the earth.

Initial orbit is almost always a low earth orbit (LEO) where it's important the orbit be circular or close to it, as an elliptical orbit would place the satellite into the earth's atmosphere during part of the orbit which would cause the orbit to degrade quickly and the satellite to burn up.

The LEO is usually used because it takes the least energy, thus the rocket can boost a larger payload.

A Hohmann transfer orbit is an elliptical orbit used to transfer between two circular orbits of different radii in the same plane. Note the phrase "between two circular orbits". So if you want the satellite in a higher orbit, rockets on the satellite itself are used to transfer it higher.

If the rocket has enough fuel or the payload is small enough, it can boost the satellite directly into the higher orbit, so a Hohmann transfer orbit is not needed. But this is less efficient as you need the energy to place the entire rocket plus satellite into the higher orbit, as opposed to the satellite only.