How fast would you have to move west to keep ahead of sunrise
in east. Assuming you had a clear pathway. With earth rotating
toward the sun.

 

Depends on the latitude. John

 

At the equator, it's close to 24,900 miles around the planet. To go that far in 24 hours means you have to move at 1,037.5 MPH. Faster at altitude, as the circumference of your path increases. In geosynchronous orbit, it's 24,900 MPH.

 

If the circumference of the earth at the equator is C, then the circumference of the earth at latitude θ (i.e., the circle created by a cut by a plane parallel to a plane containing the equator) is \(C cos \theta\). Divide that by the period of rotation (say, the sidereal day) and you have the linear velocity you desired. Add in complexities as desired (e.g., nonspherical earth, topography, etc.).

 

This speed varies based on latitude and altitude.

At 45 degrees North latitude, the required speed is 731 mph or 1176 km/h.

At 30 degrees North, the speed increases to 896 mph or 1442 km/h.

At the Equator, you would need to be going 1035 mph or 1666 km/h

This is assuming ground level. At altitude, where those speeds can be reached, the difference isn't huge, but you'd need to go a bit faster.