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Physics-

General

Easy

Question

A particle is projected with velocity $upsilon subscript 0 end subscript$ along $x minus a x i s$ . The deceleration on the particle is proportional to the square of the distance from the origin i.e., $a equals alpha x to the power of 2 end exponent.$ The distance at which the particle stops is

$\sqrt{\frac{3 υ_{0}}{2 α}}$
${(\frac{3 v_{o}}{2 α})}^{\frac{1}{3}}$
$\sqrt{\frac{3 v_{0}^{2}}{2 α}}$
${(\frac{3 v_{0}^{2}}{2 α})}^{\frac{1}{3}}$

The correct answer is: $open parentheses fraction numerator 3 v subscript 0 superscript 2 over denominator 2 alpha end fraction close parentheses to the power of 1 third end exponent$

$a equals fraction numerator d v over denominator d t end fraction equals fraction numerator d v over denominator d x end fraction fraction numerator d x over denominator d t end fraction$ $equals v fraction numerator d v over denominator d x end fraction equals negative alpha x to the power of 2 end exponent$ (given)
Þ $not stretchy integral subscript v subscript 0 end subscript end subscript superscript 0 end superscript v d v equals negative alpha not stretchy integral subscript 0 end subscript superscript S end superscript x to the power of 2 end exponent d x$ Þ $open square brackets fraction numerator v to the power of 2 end exponent over denominator 2 end fraction close square brackets subscript v subscript 0 end subscript end subscript superscript 0 end superscript equals negative alpha open square brackets fraction numerator x to the power of 3 end exponent over denominator 3 end fraction close square brackets subscript 0 end subscript superscript S end superscript$
Þ $fraction numerator v subscript 0 end subscript superscript 2 end superscript over denominator 2 end fraction equals fraction numerator alpha S to the power of 3 end exponent over denominator 3 end fraction$ Þ $S equals open parentheses fraction numerator 3 v subscript 0 end subscript superscript 2 end superscript over denominator 2 alpha end fraction close parentheses to the power of fraction numerator 1 over denominator 3 end fraction end exponent$

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