The risk I took was calculated, but man, am I bad at math

Implying that there is a privileged reference frame of the universe that you can be still relative to

lol .... time travel is possible and many species, and individuals have done it many times in the past and in the future. It's just that they keep appearing in random places in the universe in empty space.
There are hundreds, thousands, millions or billions of poor unfortunate time travellers that all appeared in empty space or embedded in rocky planets or asteroids ... or appeared next to a star and immediately just vapourized ... or just suffocated and froze to death.
And all throughout time, they all keep attempting it never knowing that it does work, but that it has unfortunate consequences.
- Speak for yourself: I for one am quite successful at traveling precisely forward at the constant speed of one second per... second. Have been all my life! :-P
  
  I've flown in aircraft at 30,000 feet .... those at ground level were passing time slower than me by millions of a fraction of a second slower than me at altitude .... when we landed, I had travelled millions of a fraction of a second faster ... I had travelled forward in time faster than everyone else
  I have seen things
- SHOCKInG REvElATioN!
  You won't believe wjat scientsts f0und in rec3nt meteor shower fragmnts!!

Aren't all time machines also space machines? Since space and time are intrinsically linked
- Yes it’s mainly user error
- Well, "space-time" machines were a bit harder to market in 3015 when Orco Corp decided to rebrand. Damn it, I shouldn't have said that!

The problem with this line of reasoning is that it assumes a universal frame of reference, which inherently doesn't exist. Motion, or lack thereof, is inherently relative.

Wouldn’t everything be relative to the gravity influence?
When you would travel through time, it would make more sense that you appear in the future in the same spot relative to the thing holding you there now, the universe has no absolute X,Y,Z coordinates.
- This is of course science fiction and up to your own interpretation. But, if you travel in time but not space, that means the earth is in a vastly different spot than it was before, even if there are no absolute coordinates to the universe. The earth travels around 140 miles per second relative to the center of the galaxy, so theoretically even a tiny time travel hop could mean you're appearing in the vacuum of space.
  
  And that galaxy rotates on itself. And that galaxy also has an elliptical movement within a cluster, which is moving within a super cluster, and so forth.
  
  Ok so you're assuming the position stays constant while time travelling. As mentioned, there is no absolute xyz, so whoever made the time machine had to choose a frame of reference in which the position would stay constant. The logical thing to do would be to take the frame of reference of the earth, and so time travel wouldn't cause any issues.

I am wondering, if your time machine was also a spaceship, when you jumped back, say, 50 years: how far would you need to travel to make it back to Earth from where you are in space if you left tomorrow (May, 29th 2025)?
- Considering the solar system is rotating around the center of the Milky Way at ~450,000 miles per hour, 50 years would have you some 197 billion miles from Earth. This is about 13x the current distance of Voyager 1 (15.6 billion miles).
  
  Shiiiiiit.
  Hope your time machine has an FTL drive. 😩

More pixels

There was a not great scifi show about time travel called seven days. They had a alien time machine that could go back in time seven days. Three trick was you had to pilot it as time went backwards chasing the earth.
The main character would often crash somewhere and have to call in to get picked up after a time jump. There was a great scene where he crashed in a civil war reenactment and had a "oh shit I went back to far" moment.

gelbanana

Unless it has enough technobabble to retrocausally gravity tether the machine in space and moved through time without somehow violating something something... Look, just reverse the polarity, it'll be fine.

I get the feeling that calculating a space shift perfectly for a long enough time jump so that one would appear exactly on the surface of planet Earth (not more that a meter or two above and definitelly not at all below) would probably be extremely hard giving the precision required and how gravitation works: theoretically every single body in the Universe affects the position of planet Earth and each other, and whilst the effect of gravity drops steeply with distance, over enough time the errors of not having calculated using all bodies add up, which for very precise positioning (not merelly "near planet Earth" but actually above but not too high a certain point in its surface) would probably get dangerous fast.
So I bet that if we ever do time travel it will only really work for really small timeframes (forget about going back a couple of million Earth orbits to the time of the dinossaurs) or it will both have a space shift and a time shift with a vehicle capable of at least limited space flight and reentry so that a "near planet Earth" error margin is acceptable in the spacial shifting part.
This is, of course, before we go into the whole time paradox discussion.
PS: I was curious so looked for and found a paper about the influences Earth's orbit. One of the interesting conclusions one can derive from it is having to take in account the collision with the asteroid that killed the dinossaurs if one wanted to travel back in time to the time of the dinossaurs with a suitable space shift to end up on the Earth's surface.

I read some time travel short story once where science found time travel was possible, but you had no control on where you go in the universe. So they went to the only place they could get to, the big bang.

Neat! So basically a free ride to an extrasolar system that's able to host life if you get really lucky. Kill two birds with one stone, what could go wrong?

V o i d b e r g

Safety not guarenteed.

You just need to phase back into time just long enough to maintain a paired momentum

Yes! Thank you. Lots of time travellers lost in the void or buried in solid objects.
Some possible ways of dealing with this:
The time travel device does the complicated math to get the time and position correct, assuming the SF writer only wants to deal with 4 dimensions.
Tunneling effect so that it's a straight point to point between the two timelines.
Get Weird. Invoke quantum entanglement to rearrange the exact same atoms that make up the time travellers' bodies in the destination timeline, because mass/energy can never be created, only borrowed, and perhaps the universe has a conservation of mass/energy rules and can't add more mass/energy from a foreign timeline without tipping the balance from a "Big Rip" end of the universe event to a "Big Crunch" event.
- My favorite patch (other than "what frame of reference are you using for this and why that one in particular, none of them are more correct than any other") is to just say gravity continues to affect you even as you're traveling in time, effectively holding you in the same place if you're on a planet or in orbit.

I always figured if the creator is smart enough to figure out how to time travel, they're smart enough to account for where things will be at that point in time.