Rectilinear Motion
Question

# The distance x covered by a body moving in a straight line in time t is given by the relation $2{\mathrm{x}}^{2}+3\mathrm{x}=\mathrm{t}$. If v is the velocity of the body at a certain instant of time, its acceleration will be

Difficult
Solution

## Differentiating $2{\mathrm{x}}^{2}+3\mathrm{x}=\mathrm{t}$ with respect to t we have $4\mathrm{x}\frac{\mathrm{dx}}{\mathrm{dt}}+\frac{3\mathrm{dx}}{\mathrm{dt}}=1$   …….(i)Now $\frac{\mathrm{dx}}{\mathrm{dt}}=\mathrm{v}$. Therefore, . Differentiating Eq. (i) with respect to time t, we have $4{\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)}^{2}+4\mathrm{x}\frac{{\mathrm{d}}^{2}\mathrm{x}}{{\mathrm{dt}}^{2}}+3\frac{{\mathrm{d}}^{2}\mathrm{x}}{{\mathrm{dt}}^{2}}=0$or  $4{\mathrm{v}}^{2}+4\mathrm{xa}+3\mathrm{a}=0$or  $\mathrm{a}=-\frac{4{\mathrm{v}}^{2}}{4\mathrm{x}+3}$  ………(ii)where $\mathrm{a}=\frac{{\mathrm{d}}^{2}\mathrm{x}}{{\mathrm{dt}}^{2}}$ is the acceleration. But $4\mathrm{x}+3=1/\mathrm{v}$. Using this in Eq. (ii) we get $\mathrm{a}=-4{\mathrm{v}}^{3}$. Hence the correct choice is (d).

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