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Question

# Velocity V of a moving point on the x-axis varies with its x coordinate as V=βx13 where β is a positive constant. Assuming the particle to start from the origin

A
Acceleration of particle is variable.
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B
Mean velocity of point averaged over time T from starting is (2β3)32T.
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C
Mean velocity of point averaged over the time taken by the point to move from the origin to the point where its x coordinate becomes xo is 2βx13o3
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D
All above statements are false.
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Solution

## The correct option is C Mean velocity of point averaged over the time taken by the point to move from the origin to the point where its x coordinate becomes xo is 2βx13o3V=βx13=dxdt ⇒∫dxx13=∫βdt ⇒32x23=βt(∵x=0 at t=0) x=(23βt)32 V=βx13 Acceleration a=dVdt=VdVdx =βx13×13βx−23=β23x−1/3 ∴ Acceleration is variable. Average velocity upto time T, vavg=Total displacementTotal time =(23βT)32T=(23β)32√T x=(23βt)32 At x=xo⇒t=3x23o2β Average velocity to xo, v=Total displacementTotal time=xo3x23o2β =2βx13o3

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