The-motion-of-a-particle-moving-along-x-axis-is-represented-by-the-equation-dv-dt-6-3v-where-v-is-in-m-s-and-t-is-in-second-If-the-particle-is-at-rest-at-t-0-then-1-The-speed-of-the-parti

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Question Number 15955 by Tinkutara last updated on 16/Jun/17

$$\mathrm{The}\mathrm{motion}\mathrm{of}\mathrm{a}\mathrm{particle}\mathrm{moving}\mathrm{along}$$$$x-\mathrm{axis}\mathrm{is}\mathrm{represented}\mathrm{by}\mathrm{the}\mathrm{equation}$$$$\frac{dv}{dt}=6-3v,\mathrm{where}v\mathrm{is}\mathrm{in}\mathrm{m}/\mathrm{s}\mathrm{and}t\mathrm{is}$$$$\mathrm{in}\mathrm{second}.\mathrm{If}\mathrm{the}\mathrm{particle}\mathrm{is}\mathrm{at}\mathrm{rest}\mathrm{at}t=$$$$0,\mathrm{then}$$$$\left(1\right)\mathrm{The}\mathrm{speed}\mathrm{of}\mathrm{the}\mathrm{particle}\mathrm{is}2\mathrm{m}/\mathrm{s}$$$$\mathrm{when}\mathrm{the}\mathrm{acceleration}\mathrm{of}\mathrm{particle}\mathrm{is}$$$$\mathrm{zero}$$$$\left(2\right)\mathrm{After}\mathrm{a}\mathrm{long}\mathrm{time}\mathrm{the}\mathrm{particle}\mathrm{moves}$$$$\mathrm{with}\mathrm{a}\mathrm{constant}\mathrm{velocity}\mathrm{of}2\mathrm{m}/\mathrm{s}$$$$\left(3\right)\mathrm{The}\mathrm{speed}\mathrm{is}0.1\mathrm{m}/\mathrm{s},\mathrm{when}\mathrm{the}$$$$\mathrm{acceleration}\mathrm{is}\mathrm{half}\mathrm{of}\mathrm{its}\mathrm{initial}\mathrm{value}$$$$\left(4\right)\mathrm{The}\mathrm{magnitude}\mathrm{of}\mathrm{final}\mathrm{acceleration}$$$$\mathrm{is}6\mathrm{m}/{\mathrm{s}}^2$$

Commented by prakash jain last updated on 16/Jun/17

$$\int \frac{dv}{6-3v}=\int dt$$$$-\frac{1}{3}\ln (6-3v)=t+C$$$$t=0,v=0$$$$-\frac{1}{3}\ln 6=C$$$$\frac{1}{3}\ln \frac{6-3v}{6}=-t$$$$\frac{6-3v}{6}=e^{-3t}\Rightarrow v=2(1-3e^{-t})\dots \left(A\right)$$$$a=0\Rightarrow \frac{dv}{dt}=0\Rightarrow 6-3v=0\Rightarrow v=2$$$$t\to \mathrm{\infty },e^{-3t}\to 0\Rightarrow v=2\left(fromA\right)$$$$t\to \mathrm{\infty },v\to z2,\frac{dv}{dt}\to 0$$$$t=0,v=0\Rightarrow a_0=6-3v=6$$$$a_t=3\Rightarrow v=1$$

Commented by Tinkutara last updated on 16/Jun/17

$$\mathrm{Thanks}\mathrm{Sir}!$$

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