Physics-
General
Easy

Question

A particle is projected with velocity v 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 negative a x to the power of 2 end exponent.The distance at which the particle stops is

  1. square root of fraction numerator 3 v subscript 0 end subscript over denominator 2 alpha end fraction end root  
  2. open parentheses fraction numerator 3 v subscript 0 end subscript over denominator 2 alpha end fraction close parentheses to the power of fraction numerator 1 over denominator 3 end fraction end exponent  
  3. square root of fraction numerator 3 v to the power of 2 end exponent subscript 0 end subscript over denominator 2 alpha end fraction end root  
  4. open parentheses fraction numerator 3 v to the power of 2 end exponent subscript 0 end subscript over denominator 2 alpha end fraction close parentheses to the power of fraction numerator 1 over denominator 3 end fraction end exponent  

The correct answer is: open parentheses fraction numerator 3 v to the power of 2 end exponent subscript 0 end subscript over denominator 2 alpha end fraction close parentheses to the power of fraction numerator 1 over denominator 3 end fraction 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 blank left square bracket G i v e n right square bracket
    rightwards double arrow blank not stretchy integral from v subscript 0 end subscript to 0 of v d v equals alpha not stretchy integral from 0 to S of x to the power of 2 end exponent d x rightwards double arrow 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 to the power of 0 end exponent 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
    rightwards double arrow 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 rightwards double arrow S equals blank open parentheses fraction numerator 3 v to the power of 2 end exponent 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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