system
Given the following post, respond as if you're commenting on Reddit:
human
During a solar eclipse if you followed the path of totality how fast would you need to be moving to stay in darkness?
gpt
It comes out to basically the speed of the Moon's orbit, which is about 1 km/s or 3600 km/h.

As the Moon orbits the Earth, it moves relative to the Sun, and that makes its shadow move relative to Earth. If the Moon moves 1° relative to the Earth-Sun line, then its shadow moves 1° from that line. So the angular speed in degrees/second of the Moon's motion relative to the Earth-Sun line is the same as the angular speed in degrees/second of the Moon's shadow.

To convert angular speed to linear speed in km/s, we convert the angular speed to radians/second, and multiply by the distance to the object. As the angular speed of the shadow (at some fixed distance, i.e. the Earth's surface) and the Moon are the same, this means that the difference in linear speed is just the ratio of the distances.

The distance from the shadow on Earth to the Sun is 1 AU. The distance from the Moon to the Sun during a solar eclipse is (1AU - 384,400km), where 384,400 km is the distance to the Moon. 1 AU is about 150 million km. So the speed of the shadow is about (moon's orbital speed)\*(150,000,000km + 384,400 km)/(150,000,000km). As 1 AU is much much bigger than the distance to the Moon, (150,000,000km + 384,400 km)/(150,000,000km) comes out to basically 1 - it's like 1.002. So the shadow on Earth moves like 0.2% faster than the Moon. The Moon goes at about 1 km/s, so the shadow goes at basically 1 km/s.