olaney
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Even if the rocket traveled "straight up", the surface of the Earth is turning, which carries the observer away from the origin of the trajectory. In this respect, it is the observer who moves horizontally with respect to the rocket rather than the other way around. To this visual effect, add whatever course correction the rocket guidance system actually makes. At the equator, the velocity of Earth's rotation is around 1000 mph, which makes the apparent horizontal component large. Most launches are planned to take advantage of the rotational velocity at the launch latitude. This is why most launch centers are located at low latitudes.