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A particle located at x=0 at time t=0, s...

A particle located at x=0 at time t=0, starts moving along the positive x-direction with a velocity 'v' which varies as `v=alphasqrt(x)` then velocity of particle varies with time as :- (`alpha` is a costant)

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(a) `v =alpha sqrtx rArr (dx)/(dt) = alpha sqrtx rArr overset x underset0 int (dx)/(sqrtx) = alpha overset t underset0 int dt rArr (x^((1)/(2)+1))/(-(1)/(2)+1)= alpha t rArr x = (alpha^(2) t^(2))/(4)`
Velocity v `=(dx)/(dt) = ( 2alpha^(2)t)/(4) = (1)/(2) alpha^(2)t`
Acceleration `a =(dv)/(dt) = (alpha^(2))/(2)`
(b) Time take to cover the first s distance
`x = (alpha^(2)t^(2))/(4) rArr s = (alpha^(2)t^(2))/(4) rArr v_(av) = (s)/(t) = (s)/(sqrt((4s)/(alpha^(2))) = (alpha sqrts)/(2)`
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