You are working under the assumption that the TARDIS moves using inertial forces. I.e. there is a force that acts on the TARDIS, moving it around like a car.
You are expecting Jack Harkness, or anyone clinging to the TARDIS, to experience these inertial forces when the TARDIS starts moving.
However, that is not provably the case. I'll list some information that can reveal how the TARDIS works.
Keep in mind that there is no direct confirmation in the lore (as far as I'm aware), this is all just a thought exercise based on what we see in the show.
Skip to chapter 4 if you just want the shorter answer.
1. Inertia inside the TARDIS (or lack thereof)
- The Doctor and his passengers (inside the TARDIS) do not experience inertia when they travel. This can be seen in pretty much every episode.
- BUT: The inside of the TARDIS is a different dimension (the door is technically speaking a portal); therefore the inertia might not carry over from the outside shell (blue box) to the inside.
- HOWEVER: When the TARDIS unexpectedly bumps into something, gets attacked by e.g. a missile, or gets handled roughly in any way (e.g. when it was picked up by a crane), the Doctor and his passengers do shake around, which means that some inertia does exist.
- Tangentially: The Doctor and his companions are never worried about standing near the open door of the TARDIS while it travels; or sitting on the edge casually. If the TARDIS were to experience inertia, that would be dangerous, because any inertia might make someone fall out of the TARDIS. Since the Doctor's companions have an active fear of dying (they are more worried about it than the Doctor) and still happily sit next to the open door; I would hazard a guess that they do not feel inertia and are therefore not afraid of falling out the door.
Most likely conclusion:
The TARDIS is capable of experiencing inertia when an outside force acts on it unexpectedly. This carries over from the outside dimension (blue box) to the inside dimension (the console).
However, when the TARDIS travels (i.e. it is not an outside force that acts on the TARDIS, but rather the TARDIS generating its own force), inertia does not seem to apply.
You could argue that the TARDIS intentionally dampens the inertia of its own propulsion force (since it knows and expects that propulsion force). It cannot dampen an unexpected force; since that (by definition) happens unexpectedly.
However, there is a much more likely (and elegant) explanation for the lack of inertia; which is supported by the TARDIS being a time machine.
2. The TARDIS is a time machine
We need to delve into the specifics as to what inertia is, before I can explain how the TARDIS does not need to use it in order to travel around.
A property of matter by which it continues in its existing state of rest or uniform motion in a straight line, unless that state is changed by an external force
The emphasis is on motion. Inertia only occurs in cases where the object is moving around in a way that the object changes its direction/speed of travel.
Example: When sitting in a car, you do not experience inertia when the car travels at a constant speed. However, when the car turns (changes direction) or accelerates/decelerates (changes speed), you will experience inertia.
Since the TARDIS can travel through space, it moves around, therefore there should be inertia; right?
But the TARDIS is also a time machine. So let's look at travelling through time rather than space.
If we only travel through time, but keep the TARDIS at a fixed location, and jump ahead in time; then the TARDIS does not move around physically. Therefore, it also experiences no inertia.
Let's look at what we know of the physical properties of time travel, according to the Whovian lore:
- The Time Vortex looks like a wormhole.
- To a passenger inside the TARDIS, when travelling in time, no inertia is experienced whatsoever.
- To an outside observer who sees the TARDIS leave, the TARDIS does not move around when it travels, it simply fades away.
- To an outside observer who sees the TARDIS arrive, the TARDIS does not move around, it simply fades into existence.
This leads to some interesting conclusions:
- There is no reason to suspect that the Tardis has physically moved between fading away and fading back into existence. It's not impossible, but there's nothing to suggest that it has been moving (in a physical sense)
- Travelling in time does not require the TARDIS to physically move.
- The TARDIS travelling through the time vortex, like we see in the Doctor Who intro, is a visual representation of travelling through time, but it is not an accurate rendition! The intro makes the TARDIS physically move around to suggest that it is travelling. But the "real world" evidence we have of the TARDIS travelling does not support this method of travel for the average time travel!
And now we get to the nitty gritty. Warning: theoretical science approaching!
3. The spacetime continuum
I'll keep this as brief as I can.
The spacetime continuum (SC from here on) is a concept that combines both space and time.
One of the more prominent theories about spacetime, is that it can be folded onto itself (much like a sheet of paper) in order to create a wormhole through which you can travel faster than normal. Please note that such a wormhole could make it possible to travel through space, through time, or both at the same time.
If you can create wormholes in spacetime, you can therefore travel through space and/or time.
The existence of the time vortex seems to support the fact that Doctor Who uses the SC; because the time vortex (when travelling) looks exactly like the spacetime continuum folded onto itself (= a wormhole, used for tavel)
If you have an X axis and a Y axis; you're working with a two dimensional graph. Now, the X axis and the Y axis are obviously very different (that's why you need both of them).
However, the method of moving position on the X axis is exactly the same as the method of changing your position on the Y axis. It's a matter of adding/subtracting the coordinate value with the amount you wish to move.
Let's move this example to a three dimensional space. Now you have an X, Y and Z axis. Three different axes, three different directions of travel.
However, the principle of moving on a given axis stays the same; regardless of whether we're talking about the X, Y or Z axis!
In other words: For both 2D and 3D spaces, if you know how to move position on one of the axes, you can use that method to move around on any of the axes.
Now, let's move this example to a four dimensional space. This is what we know as the spacetime continuum. It consists of 4 axes: X, Y, Z (= 3D space) and time (which is a single axis).
The principle is the same. If you know how to move on one axis, you can apply that method to any of the axes.
Therefore, we can assume that the method of travelling along both space (X,Y,Z) and time can be the same.
Since we have already proved that the TARDIS can travel through time without using inertial forces, the TARDIS is therefore capable of travelling through space (X,Y,Z) using that same method.
This directly proves that the TARDIS, when physically travelling, does not need to experience inertia!
- Just because you can use the same method of travel for all of the axes, doesn't mean that you have to. The TARDIS could also have a second method of travelling, e.g. only through space in a way which uses inertial. What I have proven is that the TARDIS can travel without inertial forces. Not that it always does so!
4. But how did Jack Harkness hang on to the TARDIS?
If the TARDIS travels without using inertial forces, which I have just proven is possible, then Jack Harkness doesn't risk getting thrown off. Regardless of whether he's conscious or not.
Imagine a subway line that travels at a constant speed. A passenger standing on the subway currently does not experience inertia (as the train moves at a constant speed), and therefore does not need to hold on to one of the metal bars to keep himself upright.
However, the passenger usually still holds on to the metal bar, because of safety reasons. If anything unexpected happens, they might suddenly start experiencing inertia. The reason they are already holding the metal bar before this happens is so that they can be prepared for if/when it happens.
Standing on the subway which travels at a constant speed, the passenger's hand suddenly falls asleep for a minute (which means that it's not actually holding on to the bar anymore, just resting on top of it), wakes up again after that minute, and holds on to the bar again; the passenger did not fall.
Because the subway kept moving at the same constant speed, no inertia was experienced, and therefore the passenger was able to stay upright without needing the bar to keep himself upright.
The same might be true of Jack Harkness. He might not need to hold on to the TARDIS to follow it (as there is no inertia to pull him away from the TARDIS); but he still holds on as a safety measure.
The conclusion: (this is the subway example, but substitued for Jack Harkness/the TARDIS)
Hanging onto the TARDIS which travels without inertial forces, Jack Harkness dies for a minute (which means that he's not actually holding on to the TARDIS anymore, his body is just resting against it), comes back to life again, and holds on to the TARDIS again; Jack Harkness did not get flung away from the TARDIS.
Because the TARDIS kept moving without any inertial forces being experienced, therefore Jack Harkness was able to hang on without needing to hold on tightly (so as to prevent him from being flung away).
5. A few loose ends to tie up
If the TARDIS has a second way of travelling (using inertia), why did the Doctor not use that second method of travelling so that Jack would get flung off the TARDIS?
The Doctor wasn't trying to kill Jack Harkness (it's impossible), but he also didn't want Jack to get lost in a remote part of spacetime and spend eternity lost and alone.
The Doctor was trying to deincentivize Jack from following him by travelling to places where Jack would hate going (due to him being on the outside of the TARDIS).
If Jack was mortal, the Doctor would not have done that. He knew that Jack would be fine; the travelling was only uncomfortable, not lethal.
How do you know that the TARDIS needs to follow the laws of physics?
I don't. But most things in Doctor Who, while they appear magical to us mortal humans; are based on the (in-universe) laws of physics.
The TARDIS does not behave according to our current real world laws of physics. But then again, our current laws of physics have not yet incorporated the method of time travel into its laws!
So it might be a case of our laws of physics being incomplete, rather than the TARDIS behaving irrationally. Maybe we just don't understand physics well enough yet.
I defer to the Doctor on this, as he is smarter than me. I can't refute anything he says.
I didn't want an answer this length. It was a short question. Jesus.
I was bored at work. Sorry.