The earth's circumference along the equator is 40,052km or 40,052,000m. The actual time it takes the earth to rotate 360^ (we'll say ^ is degrees) is 23 hours, 56 minutes, and 4 seconds, or 86,164 seconds [
Period of Rotation of the Earth - The Physics Factbook].
Using these two numbers you can find the earth's velocity. d=vt, so 40,052,000=v x 86,164. v then equals approximately 464.83 m/s.
Knowing the velocity, you can now determine the distance the earth travels in 24 hours, or 86,400 seconds. d=vt, so d=464.83 x 86,400 or 40,161,312m.
Using the actual circumference of the earth, you can determine the distance in meters of 1^. 40,052,000m/360^ gives you 111,255.56m per degree.
Then, dividing the distance the earth travels in 24 hours by the number of meters per degree will give you the number of degrees the earth travels in 24 hours. 40,161,312/111,255.56 gives you 360.98^ in one day, or
360^58'59".