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Maths-

General

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Question

$L minus S square root of 1 minus fraction numerator v to the power of 2 end exponent over denominator c to the power of 2 end exponent end fraction end root$
When the speed of an object approaches the speed of light, its length as seen by an observer changes. When the object is stationary relative to an observer, its length is $S$ , and when the same object is moving at speed $v$ relative to the observer, its length is $L$ . The formula above expresses $L$ in terms of $S comma v$ , and $c$ , the speed of light. Which of the following gives the speed of the object in terms of the other quantities?

$v - c \sqrt{1 - \frac{L^{2}}{S^{2}}}$
$v - c \sqrt{1 + \frac{L^{2}}{S^{2}}}$
$v = c^{2} (1 - \frac{L^{2}}{S^{2}})$
$v = c^{2} (1 + \frac{L^{2}}{S^{2}})$

The correct answer is: $v minus c square root of 1 minus fraction numerator L to the power of 2 end exponent over denominator S to the power of 2 end exponent end fraction end root$

In the given question, we have been given an equation representing the relationship between the variables L and S, v & c.
We have to convert that into a relationship between v and L, S & c.
Step-by-step solution:-
$L equals S square root of open parentheses 1 minus blank fraction numerator V to the power of 2 end exponent over denominator C to the power of 2 end exponent end fraction close parentheses end root$
$fraction numerator L over denominator S end fraction equals square root of open parentheses 1 minus blank fraction numerator V to the power of 2 end exponent over denominator C to the power of 2 end exponent end fraction close parentheses end root$
$fraction numerator L to the power of 2 end exponent over denominator S to the power of 2 end exponent end fraction equals open parentheses 1 minus fraction numerator V to the power of 2 end exponent over denominator C to the power of 2 end exponent end fraction close parentheses$ ------------------------ (Squaring both the sides)
$fraction numerator L to the power of 2 end exponent over denominator S to the power of 2 end exponent end fraction minus 1 equals fraction numerator V to the power of 2 end exponent over denominator C to the power of 2 end exponent end fraction$
$1 minus fraction numerator L to the power of 2 end exponent over denominator S to the power of 2 end exponent end fraction equals fraction numerator V to the power of 2 end exponent over denominator C to the power of 2 end exponent end fraction$ ----------------------(Multiplying both the sides by -1)
$C to the power of 2 end exponent open parentheses 1 minus fraction numerator L to the power of 2 end exponent over denominator S to the power of 2 end exponent end fraction close parentheses equals V to the power of 2 end exponent$
$C square root of open parentheses 1 minus blank fraction numerator L to the power of 2 end exponent over denominator S to the power of 2 end exponent end fraction close parentheses end root equals V$ -------------------(Taking square roots both the sides)
Final Answer:-
Option A is the correct Option.

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