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Question

A particle is dropped vertically from rest a height. The time taken by it to fall through successive distances of 1 blank m each will then be

  1. All equal, being equal to square root of 2 divided by g end root second  
  2. In the ratio of the square roots of the integers 1 comma blank 2 comma blank 3  
  3. In the ratio of the difference in the square roots of the integers i.e. square root of 1 comma blank open parentheses square root of 2 minus square root of 1 close parentheses comma open parentheses square root of 3 minus square root of 2 close parentheses comma open parentheses square root of 4 minus square root of 3 close parentheses  
  4. In the ratio of the reciprocal of the square roots of the integers i.e.,. fraction numerator 1 over denominator square root of 1 end fraction comma fraction numerator 1 over denominator square root of 2 end fraction comma fraction numerator 1 over denominator square root of 3 end fraction comma fraction numerator 1 over denominator square root of 4 end fraction  

The correct answer is: In the ratio of the difference in the square roots of the integers i.e. square root of 1 comma blank open parentheses square root of 2 minus square root of 1 close parentheses comma open parentheses square root of 3 minus square root of 2 close parentheses comma open parentheses square root of 4 minus square root of 3 close parentheses


    h equals u t plus fraction numerator 1 over denominator 2 end fraction g t to the power of 2 end exponent rightwards double arrow 1 equals 0 cross times t subscript 1 end subscript plus fraction numerator 1 over denominator 2 end fraction g t subscript 1 end subscript superscript 2 end superscript rightwards double arrow t subscript 1 end subscript equals square root of 2 divided by g end root
    Velocity after travelling 1 blank m distance
    v to the power of 2 end exponent equals u to the power of 2 end exponent plus 2 g h rightwards double arrow v to the power of 2 end exponent equals open parentheses 0 close parentheses to the power of 2 end exponent plus 2 g cross times 1 rightwards double arrow v equals square root of 2 g end root
    For second 1 blank m distance
    1 equals square root of 2 g end root cross times t subscript 2 end subscript plus fraction numerator 1 over denominator 2 end fraction g t subscript 2 end subscript superscript 2 end superscript rightwards double arrow g t subscript 2 end subscript superscript 2 end superscript plus 2 square root of 2 g end root t subscript 2 end subscript minus 2 equals 0
    t subscript 2 end subscript equals fraction numerator negative 2 square root of 2 g end root plus-or-minus square root of 8 g plus 8 g end root over denominator 2 g end fraction equals fraction numerator negative square root of 2 plus-or-minus 2 over denominator square root of g end fraction
    Taking plus v e sign t subscript 2 end subscript equals left parenthesis 2 minus square root of 2 right parenthesis divided by square root of g
    therefore fraction numerator t subscript 1 end subscript over denominator t subscript 2 end subscript end fraction equals fraction numerator square root of 2 divided by g end root over denominator left parenthesis 2 minus square root of 2 right parenthesis divided by square root of g end fraction equals fraction numerator 1 over denominator square root of 2 minus 1 end fraction and so on

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