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You might, of course, say that the time-machine partakes of the motion of the Earth, and wherever Earth goes, the time-machine goes, too. Suppose, though, we consider the Earth’s motion (with the solar system generally) around the galactic center. Its speed relative to that center is estimated to be about 220 kilometers per second. If the time-machine travels twenty-four hours into the future in one second, it travels 220 kilometers x 86,400 (the number of seconds in a day), or 19,008,000 kilometers in one second. That’s over sixty-three times the speed of light. If we don’t want to break the speed-of-light limit, then we must take not less than twenty-three minutes to travel one day forward (or backward) in time.
What’s more, I suspect that considerations of acceleration would have to be involved. The time- machine would have to accelerate to light speed and then decelerate from it, and perhaps the human body could only stand so much acceleration in the time direction. Considering that the human body has never in all its evolution accelerated at all in the time direction, the amount of acceleration it ought to be able to endure might be very little indeed, so that the time-machine would have to take considerably more than an hour to make a one-day journey-say, at a guess, twelve hours.
That would mean we could only gain half a day per day, at most, in traveling through time. Spending ten years to go twenty years into the future, would not be in the least palatable. (Can a time machine carry a life-support system of that order of magnitude?)
And, on top of that, I don’t see that having to chase after the Earth would fail to cost the usual amount of energy just because we’re doing it by way of the time dimension. Without calculating the energy, I am positive time-travel is insuperably difficult, quite apart from the theoretical considerations that make it totally impossible. So let’s eliminate it from serious consideration.
But not from science fiction! Time-travel stories are too much fun for them to be eliminated merely out of mundane considerations of impracticability, or even impossibility.