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Question

# The velocity (v) versus displacement (x) graph of a particle moving along a straight line is shown in figure. Identify the correct statement regarding the acceleration (a) of particle:

A
is constant.
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B
increases linearly with x.
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C
increases parabolically with x.
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D
None of these
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Solution

## The correct option is B increases linearly with x.Acceleration of the particle can be written as: a=dvdt=dvdxdxdt=vdvdx.............i where; dvdx= slope of velocity Vs displacement graph v= velocity of particle From the v−x graph, it is a straigh line relationship. Moreover, slope of graph i.e tanθ= +ve because θ<90∘. Hence, velocity (v) INCREASES linearly with respect to displacement x or directly proportional relation exist between v and x ∴v∝x..........ii ⇒dvdx=slope of v-x graph∴dvdx=tanθ=+ constant..........iii As a=vdvdx Combining Eq. i, ii, iii we can infer that ⇒acceleration a is varying in direct proportionality with velocity v: ∴a∝+v also from graph of v−x: v∝+x This ultimately implies that a∝x ∴ acceleration will vary in direct proportionality with displaceement (+x) and due to +ve sign of slope, It will increase. ⇒Hence, a−x graph will increase linearly.

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