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Motion In Straight Line

Question
CBSEENPH11017218

A particle moves from rest at a distance c from fixed point O with accelerationμ/x3, where μ is constant and x is the distance away from O. What will be the velocity of the particle when it is at a distance 2c from where it started?


Solution

Let at any instant the particle be at a distance x from O.
Acceleration of the particle is given by, 
      astraight v space dv over dx space equals space straight mu over straight x squared 
rightwards double arrow space straight v. dv space equals space straight mu over straight x squared dx space 

Integrating both sides, we get
space space space space space space space integral subscript 0 superscript straight v dv space equals space integral subscript straight c superscript 3 straight c end superscript straight mu over straight x squared dx space

space space space space space space space right enclose straight v squared over 2 end enclose subscript 0 superscript straight v space equals space minus right enclose straight mu over straight x end enclose subscript straight c superscript 3 straight c end superscript space

rightwards double arrow straight v squared over 2 space minus space 0 space equals space minus straight mu space open square brackets fraction numerator 1 over denominator 3 straight c end fraction minus 1 over straight c close square brackets space equals space fraction numerator 2 straight mu over denominator 3 straight c end fraction space

rightwards double arrow space space space space space space space space space space space straight v space equals space square root of fraction numerator 4 straight mu over denominator 3 straight c end fraction end root