Basically just that. Assuming the earth orbited the sun fully (360 degrees) in 24 hours, how fast would it have to travel?
It is physically impossible for Earth to orbit the sun in 24 hours and still obey the Laws of Physics. In order for that to happen, Earth's orbit would have to be so close to the sun that it would be burned ash.
However, assuming that Earth could stay in its current orbit while orbiting the sun in 24 hours, it would have to travel at a speed of 24,334,174.08 miles per hour.
Earth currently travels at an average speed of 66,622.18 miles per hour.
24 hours imply that the orbital period is 1 day (instead of the current 365.25 days). If the period is 1/365.25th, then the speed has to be 365.25 times. Current speed is about 29.78km/s or 107,228 km per hour.
If that happens to be 24 hours or 1 day the orbital speed would have to be
10879.22 km/s (39,165,188 km per hour)
In order to orbit in just 24 hours, the distance from the center of the sun would have to be 0.01957 AU = 2,927,814 km. So the orbit speed would be 766,500 km/hr = 475,281 mph. That's a bit more than seven times the speed it has now.
At that distance, the surface temperature would go up to 2000 C, which would certainly vaporize rock. But it might take many millions of years to vaporize all of the Earth.
Here is a new science discovery about a planet that orbit in just 15 hours.