An old song contains these lines: "While driving in my...

An old song contains these lines: "While driving in my Cadillac, what to my surprise; a little Nash Rambler was following me, about onethird my size." The singer of that song assumes that the Nash Rambler is driving at a similar velocity. Suppose, though, rather than actually being one-third the Cadillac's size, the proper length of the Rambler is the same as that of the Cadillac. What would be the velocity of the Rambler relative to the Cadillac for the song's observation to be accurate?

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Key Concepts

Lorentz Factor

The Lorentz factor is a key component in the equations of special relativity, defined as gamma = 1/sqrt(1 - v²/c²). It determines how much time dilates and lengths contract for objects moving relative to an observer, and it links the relative velocity to the observed changes in measurable quantities such as length.

Length Contraction

Length contraction is a phenomenon in special relativity where an object moving at a high velocity relative to an observer appears shorter along the direction of motion than its proper length (the length measured in the object's rest frame). This effect is quantified by the Lorentz factor and becomes significant as the object's speed approaches the speed of light.

Special Relativity

Special relativity, formulated by Einstein, is the framework that describes the physics of objects moving at speeds close to the speed of light. It introduces the idea that measurements of space and time depend on the relative motion of observers, leading to phenomena such as time dilation and length contraction.

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