Physics-
General
Easy

Question

A stone dropped from a balloon which is at a height h comma reaches the ground after t second. From the same balloon, if two stones are thrown, one upwards and the other downwards, with the same velocity u and they reach the ground after t subscript 1 end subscript a n d blank t subscript 2 end subscript second respectively, then

  1. t equals t subscript 1 end subscript minus t subscript 2 end subscript  
  2. t equals fraction numerator t subscript 1 end subscript plus t subscript 2 end subscript over denominator 2 end fraction  
  3. t equals square root of t subscript 1 end subscript t subscript 2 end subscript end root  
  4. t equals square root of t subscript 1 end subscript superscript 2 end superscript minus t subscript 2 end subscript superscript 2 end superscript end root  

The correct answer is: t equals square root of t subscript 1 end subscript t subscript 2 end subscript end root


    From equation of motion, we have
    s equals u t plus fraction numerator 1 over denominator 2 end fraction g t to the power of 2 end exponent
    Where, u is initial velocity, g the acceleration due to gravity and t the time.
    For upward motion
    h equals negative u t subscript 1 end subscript minus fraction numerator 1 over denominator 2 end fraction g t subscript 1 end subscript superscript 2 end superscript blank open parentheses i close parentheses
    for downward motion
    blank h equals negative u t subscript 2 end subscript plus fraction numerator 1 over denominator 2 end fraction g t subscript 2 end subscript superscript 2 end superscript blank open parentheses i i close parentheses
    multiplying Eq. (i) by t subscript 2 end subscript and Eq. (ii) by t subscript 1 end subscript and subtracting Eq. (ii) by Eq. (i), we get
    blank h open parentheses t subscript 2 end subscript minus t subscript 1 end subscript close parentheses equals fraction numerator 1 over denominator 2 end fraction g t subscript 1 end subscript t subscript 2 end subscript open parentheses t subscript 2 end subscript minus t subscript 1 end subscript close parentheses
    h equals fraction numerator 1 over denominator 2 end fraction blank g t subscript 1 end subscript t subscript 2 end subscript blank open parentheses i i i close parentheses
    When stone is dropped from rest u equals 0 comma reaches the ground in t second.
    therefore blank h equals fraction numerator 1 over denominator 2 end fraction g t to the power of 2 end exponent blank open parentheses i v close parentheses
    Equating Eqs. (iii) and (iv), we get
    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 blank t subscript 1 end subscript t subscript 2 end subscript
    rightwards double arrow blank t to the power of 2 end exponent equals t subscript 1 end subscript t subscript 2 end subscript blank rightwards double arrow t equals square root of t subscript 1 end subscript t subscript 2 end subscript end root

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