How fast would you need to be travelling on the ground to keep up with the sun?
About 1000 MPH
It depends on your latitude and the time of year. At the equator at the equinox, it’d be about 1000 miles per hour. The speed would be slower as you go north or south.
Circumference at the equator is about 40,000 kilometers.
Divide this by 24 hours: 1,667 kilometers per hour. Or, indeed, 1,036 MPH.
The speed for any particular latitude is *V = (40000 kph /24)cos(theta), where theta is the latitude north or south of the equator.
And of course, at the Poles, you wouldn’t have to move. 
Not true. Since the Earth is tilted at 23.5 degrees to the plane of the solar system, the poles are in darkness for a good portion of the year. In the effort to keep up with the Sun, one would not be following a line of latitude, but rather a great circle parallel to the plane of the solar system. An implied condition of the OP is that one can see the sun from the ground at all times.
Okay, so, correct me on this scenario if I’m wrong:
You start at the North Pole. You sit motionless for, say, 160 days, while the sun spins around you. (I’m disregarding the Copernican revolution. Call the Earth the stable frame of reference, given the question is “how fast would you have to go,” i.e. compared to the stationary Earth.)
Eventually, the sun dips to the horizon. Right before it disappears from view, you begin an initially slow but rapidly accelerating spiral away from the Pole. Your spiral takes you further and further south for a week or so. You pass through the Equator at around 1000 mph, at which point you begin to decelerate over the next week or so.
Finally, you arrive at the South Pole, where you can sit motionless for another 160 days, while the sun spins around you.
So. What are the exact numbers? My guess of 160 days at each pole is just that, a wild guess. I worked backward from that guess to offer a vague “couple of weeks” spiral transit down the globe.
(Kudos to the OP. This is a really weird, but interesting question…)
If you wanted to keep the Sun directly overheard, the slowest you could go (ie, how fast you would have to go) would be on one of the Solstices, at one of the Tropics. Your speed would be about 951 MPH. And although you yourself would stay in the plane of the Solar System, the path you traced out on Earth would more or less be a line of latitude.
Cervaise, if you’re asking whether that scenario would keep you in the light at all times, I think the answer’s yes. I was thinking, if you wanted to stay in the sunlight for one full year, and you wanted to wind up back where you started, what’s the shortest distance you could travel? And I think the answer is one Earth circumference, or about 25,000 miles (for an average speed of 2.85 MPH). It’s the same idea as what you posted, but you’d have to zip from one pole to the other really really fast, in order to go directly from one to the other without seeing the Sun set.
If you wanted to keep the Sun directly overheard, the slowest you could go (ie, how fast you would have to go) would be on one of the Solstices, at one of the Tropics. Your speed would be about 951 MPH. And although you yourself would stay in the plane of the Solar System, the path you traced out on Earth would more or less be a line of latitude.
Cervaise, if you’re asking whether that scenario would keep you in the light at all times, I think the answer’s yes. I was thinking, if you wanted to stay in the sunlight for one full year, and you wanted to wind up back where you started, what’s the shortest distance you could travel? And I think the answer is one Earth circumference, or about 25,000 miles (for an average speed of 2.85 MPH). It’s the same idea as what you posted, but you’d have to zip from one pole to the other really really fast, in order to go directly from one to the other without seeing the Sun set.
sitting on a pole for 1/2 a year does not require a quick jaunt to the other pole, you just have to circle further twards the equator to keep in the daylight reagon. perhaps at the artic circle.
I disagree. For half of the year, there is no point in the Arctic Circle that remains in daylight. If you stayed in the Arctic Circle, you’d have to keep walking around the pole to stay in the sunlight, making larger and larger circles, until you reached the Winter Solstice, when you’d be walking along the Arctic Circle itself. I estimate that this gives a path length of about 1.17 Million miles, for an averge speed of 133 MPH.