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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 options are
A Acceleration of particle is variable.
B Mean velocity of point averaged over time T from starting is (2β3)32T.
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
V=βx13=dxdt
dxx13=βdt
32x23=βt(x=0 at t=0)
x=(23βt)32
V=βx13
Acceleration a=dVdt=VdVdx
=βx13×13βx23=β23x1/3
∴ Acceleration is variable.
Average velocity upto time T,
vavg=Total displacementTotal time
=(23βT)32T=(23β)32T
x=(23βt)32
At x=xot=3x23o2β
Average velocity to xo,
v=Total displacementTotal time=xo3x23o2β
=2βx13o3

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