 All our speed so far, except for the last one, was relative to the surface of the Earth. In the time before we knew the Earth was spinning on its axis once a day, and rotating around the Sun once a year, everyone thought that the Universe had a preferred frame of reference against which all other speeds could be measured. That preferred frame was the Earth, the center of the Universe. But once Galileo spotted the moons around Jupiter, another approach was needed. We'll use a train example. Let's measure the speed of the person walking on the train. We'll use the same measuring technique we used for the snail. The person on the train sets his clock to zero, marks his starting spot, walks down the car, stops the clock, and marks the second point. Now he just measures the length of the line and divides by the time. In this example, he went 3 meters in 5 seconds for a speed of 0.6 meters per second. Now picture the train car moving slowly to the right at 2 kilometers per hour, or 0.56 meters per second. This is the speed as measured by a person on the ground. We then repeat the measurement for the observer on the ground who is watching the train go by. He sets his clock to zero at the same time the rider on the train does. He marks the rider's starting spot. He watches the rider move down the moving car. He stops the clock when the rider does and marks the second point. Now using the same process, he just measures the length of the line and divides by the time. In this example, the rider went 5.8 meters in the same 5 seconds for a speed of 1.16 meters per second. Who was correct? Is he moving at 0.6 meters per second, or almost double that speed, at 1.16 meters per second? In the old system before Galileo, you could argue that the observer on the ground was correct, but in the actual world of equal reference frames, both are correct. In fact, we could have done it from the point of view of the train instead of the person on the ground. In that case, it is the person on the ground that is moving at 0.56 meters per second to the left, instead of the train moving to the right. If we put this on our space time graph, we see the train moving as the inertial frame velocity v' and the person walking with the velocity 1.1 meters per second. Now just rotate the velocity lines to make the train standing still. This turns it into the space time graph for the train's frame of reference. Here we see that the ground is moving backwards, at 0.56 meters per second, and the person on the train is moving at 0.6 meters per second. With this in mind, to be completely accurate, the statement needs to be worded as the person on the train is moving at 0.6 meters per second with respect to the train. And the person on the train is moving at 1.16 meters per second with respect to the ground. You can see that we are simply adding the speed of the train to the speed of the person with respect to the train. This is the Galilean transformation between two reference frames moving at constant speed with respect to each other. These are called inertial frames because they are not experiencing any acceleration. In this model, time flows at the same rate in all inertial reference frames and all motion is relative. The Galilean transformations give us the equations for converting from one frame to another. Let's look at another example. Here the train is moving faster at 25 meters per second. The person on the train kicks a ball in the direction of the train movement and measures its speed at 10 meters per second. The person on the ground would add this to the speed of the train and gets 35 meters per second. Now if the person kicks the ball in the opposite direction, the person on the ground would subtract the speed of the ball from the speed of the train. He would see it moving at 15 meters per second. Here's another example that illustrates that it doesn't matter what is moving. Suppose the person on the train kicks a water container initiating a sound wave in the water moving in the direction of the train. He would measure the speed of sound in water as being the same when he kicks it forward and when he kicks it backward. The speed of sound in water is around 1484 meters per second. The person on the ground would measure the forward moving wave at 25 meters per second faster than that and he would measure the backward moving wave at 25 meters per second slower than that. It followed that if it were a light bulb that the person on the train turned on, he would see the light moving in the direction of the train and the light moving in the opposite direction of the train to be the same speed of light. But the person on the ground would measure the light moving with the train at 25 meters per second faster than that and the speed of light traveling against the movement of the train at 25 meters per second slower than that. This view stood the test of time from Galileo until the mid-1800s because no one could measure the speed of light and no one had instruments sensitive enough to measure these small differences in the speed of light.