Physics-
General
Easy

Question

The retardation experienced by a moving motor boat, after its engine is cut off, is given by
fraction numerator d u over denominator d t end fraction equals negative k v to the power of 3 end exponent commawherek is a constant.
If v subscript 0 end subscriptis the magnitude of the velocity at cut-off, the magnitude of the velocity at time t after the cut-off is

  1. v subscript 0 end subscript  
  2. fraction numerator v subscript 0 end subscript over denominator 2 end fraction  
  3. v subscript 0 end subscript e to the power of negative k t end exponent  
  4. fraction numerator v subscript 0 end subscript over denominator square root of 2 v subscript 0 end subscript superscript 2 end superscript k t plus 1 end root end fraction  

The correct answer is: fraction numerator v subscript 0 end subscript over denominator square root of 2 v subscript 0 end subscript superscript 2 end superscript k t plus 1 end root end fraction


    not stretchy integral subscript v subscript 0 end subscript end subscript superscript v end superscript fraction numerator d v over denominator k v to the power of 3 end exponent end fraction equals not stretchy integral subscript 0 end subscript superscript t end superscript d t
    negative fraction numerator 1 over denominator k end fraction not stretchy integral from v subscript 0 end subscript to v of v to the power of negative 3 end exponent d v equals t or negative fraction numerator 1 over denominator k end fraction open vertical bar fraction numerator v to the power of negative 3 plus 1 end exponent over denominator negative 3 plus 1 end fraction close vertical bar subscript v subscript 0 end subscript end subscript superscript v end superscript equals t
    or fraction numerator 1 over denominator 2 k end fraction open square brackets fraction numerator 1 over denominator v to the power of 2 end exponent end fraction minus fraction numerator 1 over denominator v subscript 0 end subscript superscript 2 end superscript end fraction close square brackets equals t or fraction numerator 1 over denominator v to the power of 2 end exponent end fraction minus fraction numerator 1 over denominator v subscript 0 end subscript superscript 2 end superscript end fraction equals 2 k t
    or fraction numerator 1 over denominator v to the power of 2 end exponent end fraction equals fraction numerator 1 over denominator v subscript 0 end subscript superscript 2 end superscript end fraction plus 2 k t or fraction numerator 1 over denominator v to the power of 2 end exponent end fraction equals fraction numerator 1 plus 2 v subscript 0 end subscript superscript 2 end superscript k t over denominator v subscript 0 end subscript superscript 2 end superscript end fraction
    or v equals fraction numerator v subscript 0 end subscript over denominator square root of 2 v subscript 0 end subscript superscript 2 end superscript k t plus 1 end root end fraction

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