Does momentum conservation not apply to transporter operation in the Star Trek Universe? From what we have been shown, apparently not.

Given the transporters already violate so many of our current laws of physics including:

mass-to-energy conversion and back again,

quantum data storage and information manipulation,

not to mention manipulation of subspace to move the matter stream from one location to another,

the manipulation of an aspect of matter as simple as momentum from a falling object should be relatively easy to reduce a percentage of that in transport, possibly bleeding it off into subspace at the completion of the transport and reconstruction cycle.

Okay, it's not so simple but let's reverse engineer it and see if we can make it make sense...

What we know

The Law of Conservation of Momentum does indicate that within a closed system, momentum is conserved.

Two Starfleet officers are plunging to the deaths on the planet below.

As such, the closed system is composed of the officers, the planet and their momentum gained in freefall.

The conservation takes place when the potential energy of their fall is converted to kinetic energy upon landing, creating a nice cloud of dust, some heat from impact, and the squashing of all of their bones and organs to a nice jelly.

Did anyone else think this was a bad idea?

HOWEVER

Add a transporter beam which adds energy to this closed system, converts matter into energy and dematerializes the matter into a subspace confinement beam and then stores that stream of matter as a signal in a pattern buffer and biofilter.

IS THAT MATTER MOVING ANY LONGER?

Not at all. So any momentum the matter has after it is reconstituted should be able to be manipulated as just another variable in the transporter resequencing, meaning it can be modified or even removed if necessary.

I mention this because we have seen the transporter grab fighter pilots from the seats of their moving planes coming right at the Enterprise. (Tomorrow is Yesterday - TOS, Episode 1x21)

We have also seen the crew of the Galileo 7 being transported back to the Enterprise as the shuttlecraft they were on was re-entering the planet's atmosphere. (The Galileo Seven - TOS, Episode 1x13)

For that matter, anytime anyone is transported to any other location, there is almost always a movement differential that must be accounted for. Ships in orbit transporting anyone to the ground have to account for the rotational energy of the planet. I doubt seriously orbiting ships are forced to match course and speed to compensate for planetary rotation...

If momentum was conserved, they would splat against the wall of the transporter room upon reintegration, so it must be possible to treat momentum as just another variable to be manipulated mid-transport.

As to the Portal Gun

The level of technological sophistication between the transporter and the portal gun is the same level of difference between a bonfire and a tokamak reactor.

The portal gun opened a dimensional rift between two locations able to be bridged almost like a stable wormhole between two points.

Since the wormhole generated is still just an open doorway from one location to the next, all the gravitational, physical and energy requirements are unchanged from one side of the portal to the other, things falling between the two locations for example maintain their momentum and trajectory.

This means all normal physical rules apply since the only variation in the physics is the doorway between the two locations in space-time. A fall from a great height through one portal in the floor, while another is placed on a nearby wall will transform the energy of the fall into a corresponding distance value depending on the speed of exit from the wall portal.

There is no conversion of matter or energy, nor is any form of computational interface involved. Anything moving between the two boundaries is unchanged in any way. These two technologies are vastly different in their physics and their interaction with physical matter.

The Portal Gun simply allows matter to be moved from one location to another without any change in the state of that matter. The transporter is able to affect the state, position, relationship, momentum and energy condition of the transported material reconstituting the matter from an intermediate energy state.