Physics-
General
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Question

The acceleration of a particle is increasing linearly with time t blankas b t.The particle starts from the origin with an initial velocity v subscript 0 end subscript. The distance travelled by the particle in time t will be

  1. v subscript 0 end subscript t plus fraction numerator 1 over denominator 3 end fraction b t to the power of 2 end exponent  
  2. v subscript 0 end subscript t plus fraction numerator 1 over denominator 3 end fraction b t to the power of 3 end exponent  
  3. v subscript 0 end subscript t plus fraction numerator 1 over denominator 6 end fraction b t to the power of 3 end exponent  
  4. v subscript 0 end subscript t plus fraction numerator 1 over denominator 2 end fraction b t to the power of 2 end exponent  

The correct answer is: v subscript 0 end subscript t plus fraction numerator 1 over denominator 6 end fraction b t to the power of 3 end exponent


    fraction numerator d v over denominator d t end fraction equals b t rightwards double arrow d v equals b t blank d t rightwards double arrow v equals fraction numerator b t to the power of 3 end exponent over denominator 2 end fraction plus K subscript 1 end subscript
    At t equals 0 comma v equals v subscript 0 end subscript rightwards double arrow K subscript 1 end subscript equals v subscript 0 end subscript
    We get v equals fraction numerator 1 over denominator 2 end fraction b t to the power of 2 end exponent plus v subscript 0 end subscript
    Again fraction numerator d x over denominator d t end fraction equals fraction numerator 1 over denominator 2 end fraction b t to the power of 2 end exponent plus v subscript 0 end subscript
    rightwards double arrow x equals fraction numerator 1 over denominator 2 end fraction fraction numerator b t to the power of 3 end exponent over denominator 3 end fraction plus v subscript 0 end subscript t plus K subscript 2 end subscript
    At t equals 0 comma x equals 0 rightwards double arrow K subscript 2 end subscript equals 0
    therefore x equals fraction numerator 1 over denominator 6 end fraction b t to the power of 3 end exponent plus v subscript 0 end subscript t

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