An object may be regarded to be at rest or in motion depending on… - Vidyadip

Question

An object may be regarded to be at rest or in motion depending on the frame of reference chosen to view the object. Because of length contraction it would mean that the same rod may have two different lengths depending on the state of the observer. Is this true?

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Answer

Yes, it is true. If a rod is moving at a certain speed v, its contracted length is given as $\text{l}=\text{l}_0\sqrt{1-\frac{\text{v}^2}{\text{c}^2}},\text{v}<\text{c}$ where $\text{l}_0$ is the length of the rod at rest, i.e. the length varies when measured from different frames. If the observer and the rod are moving with the same speed v in the same direction, then length of the rod l is equal to $\text{l}_0$, Also, if they are moving with the same speed v in the opposite direction, the measured length of the rod measured is given by:$\text{l}=\text{l}_0\sqrt{1-\frac{(\text{v}-(-\text{v})^2}{\text{c}^2}}=\text{l}_0\sqrt{1-\frac{4\text{v}^2}{\text{c}^2}}$

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