Nothing! Time does not depend on the rotation of the Earth. Yes it is true that we set our clocks to the Earths rotation but that's just a matter of convenience. Time itself is more fundamental.

The Earth is not moving any near fast enough for the effects of time dilation and special relativity to have an effect.

Objects in the Solar System generally speaking are moving with respect to each-other at tens of kilometers per second, and the speed of light is three-hundred thousand kilometers per second. You need to be going a significant fraction of the speed of light for time dilation to be important. It is of course an effect at any speed, but it is (almost) not measurable.

What would happen if you were going fast enough for time dilation to be important? You would not notice anything until you looked out the window of your spaceship. Then you would notice that everything else would be moving more slowly, compared to you. That is why the theory is called relativity: it is how fast things are going relative to you that matters. There is no fundamental state of "not moving", and the speed of light in all directions is the same, no matter how you are moving.

How can things be slow on Earth and not for you, when I (on Earth) think that you are going slowly and I am not? This is called the Twin Paradox, and it stems from the fact that in order for us to truly compare how fast time is traveling for each of us, we must be not moving with respect to each-other, and then be moving with respect to each-other. This means that acceleration must take place, since one of us has to go from one state of motion to another. Special relativity only functions if there is no acceleration (actually, if you break the acceleration into discontinuous jumps, you can still do it, but the algebra gets really intense). For real-world problems of accelerating systems, you need general relativity.I do not have the understanding of mathematics to be able to explain general relativity to you.

Well, the short answer is that nothing much would happen. The Earth rotates at a rate about 1000 miles per hour which is faster than the speed of sound but not really by a lot. So traveling this speed would be like flying in a fighter jet which military pilots do all the time. By the way, the space shuttle travels at about 17,000 miles per hour. The Earth travels around the Sun at a faster rate - about 67,000 miles per hour by my calculations - but still much slower than the speed of light which is 186,000 miles per second!

This is a great question. If you were traveling the same speed as the earth and heading in the opposite direction, then for you, day and night would cease to exist as the 24-hour cycles that we are familiar with. Instead, you would remain in the same position relative to the sun and experience the same time of day (or night) perpetually without ever seeing the repeated cycle of sunrise, day, sunset, and night.

However, time is a different concept than day and night. We think of time only in terms of day and night because the two are related for anybody living on Earth and turning with the planet. But this relationship between time and the day and night caused by Earth's rotation ends as soon as you leave the planet's rotational frame of reference.

As an example,have you ever flown in a plane from east to west, especially at sunset? It appears that the sunset almost freezes and takes forever to finish, almost as if time were to slow down. This is because your position relative to the sun is changing very slowly because you are traveling in the opposite direction of earth's rotation and effectively "chasing the sun". However, time does not slow down either for you or the people on Earth who the pilots are communicating with. Time would not stop as it always continues in the same manner.

However, whenever you speed up relative to someone else, you would experience time at a negligibly different rate than people below on Earth, due to Einstein's theory of special relativity. This concept is called "time dilation" and occurs because the speed of light is constant regardless of your frame of reference.

In order for the speed of an external object (light) to be the same for two things moving at different velocities relative to that object,time must slow down for the object that is moving faster away from the source of the light and time must speed up for the object that is moving faster towards the source of light. This is because velocity equals distance divided by time. When you move away from a source of light, you are essentially increasing the distance that the light travels to get to you. In order for the velocity of light to remain constant, you must also increase the number of seconds that it takes the light to travel that distance. Conversely, if you move towards a source of light, you are decreasing the distance the light travels to get to you. To keep the velocity of light constant from your frame of reference, you must also decrease the number of seconds that it takes the light to travel that distance.