Imagine you attend the matinee and evening performances of the same play. At the matinee performance, the role of the King is played by Don and the role of the Court Jester is played by Brad. In the evening performance, they swap places: the role of the King is played by Brad and the role of the Court Jester is played by Don. Regardless of who plays each role, you are able to watch, understand, and enjoy both performances.

The same type of swap occurs in mathematic and physics. Revisiting the example of the bus from Tutorial six, the starting position of the bus is represented by the variable x and the ending position of the bus is represented by the variable x’. However, this choice is arbitrary and we could have just as easily said that the starting position of the bus is x’ and the ending position of the bus is x. So, if the front of the bus begins at x’ and travels at velocity v for t seconds, it will arrive at position x represented by the equation:

x = x’ + vt

This relationship is illustrated as:

Generally, we use this equation to determine a future position: Where is the bus after t seconds? But, we can also use this equation to determine a past position. If we know where the bus is now, represented by x, we can ask where was the bus located t seconds ago, which is represented by x’. This is illustrated as:

The answer to the question: If the bus is now located as x, where was the bus located t seconds ago, is found by rearranging the expressions to produce the equation:

x’ = x – vt

As you can see, the translation equation is highly versatile and forms the foundation of Modern Mechanics and classical mechanics. Interestingly, while the translation equation in this second form is used in Einstein’s relativity theory, its use is universally overlooked. We’ll examine this oversight in a later tutorial. But rest assured, regardless of the theory, the translation equation is always at play.

Fact: In Modern Mechanics, classical mechanics, and relativity theory, a moving system always moves according to the translation transformation.

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