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Question: A stick of length one meter is moving with a velocity of \(2.7\times {{10}^{8}}m{{s}^{-1}}\). What i...

A stick of length one meter is moving with a velocity of 2.7×108ms−12.7\times {{10}^{8}}m{{s}^{-1}}. What is the apparent length of the stick?
a)10m b)0.22m c)0.44m d)2.4m \begin{aligned} & a)10m \\\ & b)0.22m \\\ & c)0.44m \\\ & d)2.4m \\\ \end{aligned}

Explanation

Solution

The velocity of the object decreases when an object travels with a high velocity. Using the special theory of relativity, the new length that will appear to us can be calculated using the special theory of relativity. Einstein developed this formula during his work on general and special theories of relativity.

Formula used: l=l01−v2c2l={{l}_{0}}\sqrt{1-\dfrac{{{v}^{2}}}{{{c}^{2}}}}

Complete step by step answer:
Let is write down the given values below,
lo=1m ⇒v=2.7×108ms−1 ⇒c=3×108ms−1 \begin{aligned} & {{l}_{o}}=1m \\\ & \Rightarrow v=2.7\times {{10}^{8}}m{{s}^{-1}} \\\ & \Rightarrow c=3\times {{10}^{8}}m{{s}^{-1}} \\\ \end{aligned}
Length contraction in special theory relativity is given by,
l=l01−v2c2l={{l}_{0}}\sqrt{1-\dfrac{{{v}^{2}}}{{{c}^{2}}}}
Substituting the above values, we get,
l=l01−v2c2 l=1×1−(2.7)29 l=0.435m \begin{aligned} & l={{l}_{0}}\sqrt{1-\dfrac{{{v}^{2}}}{{{c}^{2}}}} \\\ & l=1\times \sqrt{1-\dfrac{{{(2.7)}^{2}}}{9}} \\\ & l=0.435m \\\ \end{aligned}

So, the correct answer is “Option C”.

Additional Information: Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is length as measured in the object's own frame. It can also be described as the Lorentz contraction or Lorentz Fitzgerald contraction. This length contradiction is usually noticeable at a substantial fraction of the speed of light. Length contraction is only in the direction in which the body is travelling. For standard objects, this effect is negligible at everyday speeds, and can be ignored for all regular purposes, only becoming significant as the object approaches the speed of light of light relative to the observer.

Note: The length contradiction in the above question is countable because the speed at which the body is travelling is almost equal to or very close to the speed of light. At normal speeds, our eye is able to capture the frames of the object easily which makes the length contradiction very much negligible.