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Question

A car starts from rest and moves with uniform acceleration a on a straight road from time t equals 0 blankto t equals T. After that, a constant deceleration brings it to rest. In this process the average speed of the car is

  1. fraction numerator a T over denominator 4 end fraction  
  2. fraction numerator 3 a T over denominator 2 end fraction  
  3. fraction numerator a T over denominator 2 end fraction  
  4. a T  

The correct answer is: fraction numerator a T over denominator 2 end fraction


    For first part,
    u equals 0 comma blank t equals T and acceleration equals a
    therefore v equals 0 plus a T equals a T and S subscript 1 end subscript equals 0 plus fraction numerator 1 over denominator 2 end fraction a T to the power of 2 end exponent equals fraction numerator 1 over denominator 2 end fraction a T to the power of 2 end exponent
    For Second part,
    u equals a T comma retardation equals a subscript 1 end subscript comma blank v equals 0 and time taken equals T subscript 1 end subscript (let)
    therefore 0 equals u minus a subscript 1 end subscript T subscript 1 end subscript rightwards double arrow a T equals a subscript 1 end subscript T subscript 1 end subscript
    And from v to the power of 2 end exponent equals u to the power of 2 end exponent minus 2 a S subscript 2 end subscript rightwards double arrow S subscript 2 end subscript equals fraction numerator u to the power of 2 end exponent over denominator 2 a subscript 1 end subscript end fraction equals fraction numerator 1 over denominator 2 end fraction fraction numerator a to the power of 2 end exponent T to the power of 2 end exponent over denominator a subscript 1 end subscript end fraction
    S subscript 2 end subscript equals fraction numerator 1 over denominator 2 end fraction a T cross times T subscript 1 end subscript open parentheses A s a subscript 1 end subscript equals fraction numerator a T over denominator T subscript 1 end subscript end fraction close parentheses
    therefore v subscript a v end subscript equals fraction numerator S subscript 1 end subscript plus S subscript 2 end subscript over denominator T plus T subscript 1 end subscript end fraction equals fraction numerator fraction numerator 1 over denominator 2 end fraction a T to the power of 2 end exponent plus fraction numerator 1 over denominator 2 end fraction a T cross times T subscript 1 end subscript over denominator T plus T subscript 1 end subscript end fraction
    equals fraction numerator fraction numerator 1 over denominator 2 end fraction a T left parenthesis T plus T subscript 1 end subscript right parenthesis over denominator T plus T subscript 1 end subscript end fraction equals fraction numerator 1 over denominator 2 end fraction a T

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