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Question

A body A is thrown up vertically from the ground with a velocity V subscript 0 end subscript and another body B is simultaneously dropped from a height H. They meet at a height fraction numerator H over denominator 2 end fraction if V subscript 0 end subscript is equal to

  1. square root of 2 g H end root  
  2. square root of g H end root  
  3. fraction numerator 1 over denominator 2 end fraction square root of g H end root  
  4. square root of fraction numerator 2 g over denominator H end fraction end root  

The correct answer is: square root of g H end root


    Let the two bodies A and B respectively meet at a time t, at a height fraction numerator H over denominator 2 end fraction from the ground

    Using S equals u t plus fraction numerator 1 over denominator 2 end fraction a t to the power of 2 end exponent
    For a body A comma blank u equals V subscript 0 end subscript comma blank a equals negative g comma blank S equals fraction numerator H over denominator 2 end fraction
    therefore fraction numerator H over denominator 2 end fraction equals V subscript 0 end subscript t minus fraction numerator 1 over denominator 2 end fraction g t to the power of 2 end exponent(i)
    For body B comma blank u equals 0 comma blank a equals plus g comma blank S equals fraction numerator H over denominator 2 end fraction
    therefore fraction numerator H over denominator 2 end fraction equals fraction numerator 1 over denominator 2 end fraction g t to the power of 2 end exponent (ii)
    Equating equations (i) and (ii), we get
    V subscript 0 end subscript t minus fraction numerator 1 over denominator 2 end fraction g t to the power of 2 end exponent equals fraction numerator 1 over denominator 2 end fraction g t to the power of 2 end exponent rightwards double arrow V subscript 0 end subscript t equals g t to the power of 2 end exponent or t equals fraction numerator V subscript 0 end subscript over denominator g end fraction
    Substituting the value of t in equation (i), we get
    fraction numerator H over denominator 2 end fraction equals V subscript 0 end subscript cross times open parentheses fraction numerator V subscript 0 end subscript over denominator g end fraction close parentheses minus fraction numerator 1 over denominator 2 end fraction g open parentheses fraction numerator V subscript 0 end subscript over denominator g end fraction close parentheses to the power of 2 end exponent equals fraction numerator V subscript 0 end subscript superscript 2 end superscript over denominator g end fraction minus fraction numerator 1 over denominator 2 end fraction fraction numerator V subscript 0 end subscript superscript 2 end superscript over denominator g end fraction
    fraction numerator H over denominator 2 end fraction equals fraction numerator 1 over denominator 2 end fraction fraction numerator V subscript 0 end subscript superscript 2 end superscript over denominator g end fraction or V subscript 0 end subscript superscript 2 end superscript equals g H rightwards double arrow V subscript 0 end subscript equals square root of g H end root

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