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Question Number 15039 by Tinkutara last updated on 07/Jun/17

$$\mathrm{The}\mathrm{acceleration}\mathrm{of}\mathrm{an}\mathrm{object}\mathrm{is}\mathrm{given}$$$$\mathrm{by}a\left(t\right)=\cos \left(\pi t\right){\mathrm{ms}}^{-2}\mathrm{and}\mathrm{its}\mathrm{velocity}$$$$\mathrm{at}\mathrm{time}t=0\mathrm{is}\frac{1}{2\pi }\mathrm{m}/\mathrm{s}\mathrm{at}\mathrm{origin}.\mathrm{Its}$$$$\mathrm{velocity}\mathrm{at}t=\frac{3}{2}\mathrm{s}\mathrm{is}?$$$$\mathrm{The}{\mathrm{object}}^{\prime }\mathrm{s}\mathrm{position}\mathrm{at}t=\frac{3}{2}\mathrm{s}\mathrm{is}?$$

Answered by ajfour last updated on 07/Jun/17

$$v-\frac{1}{2\pi }=\frac{\sin \pi t}{\pi }$$$$v{\mid }_{t=3/2}=\frac{1}{2\pi }-\frac{1}{\pi }=-\frac{1}{2\pi }m/s$$$$x=\frac{t}{2\pi }-\frac{(\cos \pi t-1)}{{\pi }^2}$$$$x{\mid }_{t=3/2}=(\frac{3}{4\pi }+\frac{1}{{\pi }^2})m.$$

Commented by Tinkutara last updated on 07/Jun/17

$$\mathrm{But}\mathrm{answer}\mathrm{given}\mathrm{is}\left(\mathrm{i}\right):\frac{3}{2\pi }\mathrm{m}/\mathrm{s}$$$$\left(\mathrm{ii}\right):\frac{3}{4\pi }+\frac{1}{{\pi }^2}\mathrm{m}$$

Commented by mrW1 last updated on 07/Jun/17

$$\mathrm{the}\mathrm{answer}\mathrm{given}\mathrm{in}\mathrm{your}\mathrm{book}\mathrm{is}\mathrm{wrong}.$$

Commented by Tinkutara last updated on 07/Jun/17

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

Answered by mrW1 last updated on 07/Jun/17

$$\mathrm{v}\left(\mathrm{t}\right)=\int \mathrm{a}\left(\mathrm{t}\right)\mathrm{dt}=\int \cos \left(\pi \mathrm{t}\right)\mathrm{dt}=\frac{\sin \left(\pi \mathrm{t}\right)}{\pi }+\mathrm{C}$$$$\mathrm{since}\mathrm{v}=\frac{1}{2\pi }\mathrm{at}\mathrm{t}=0\Rightarrow \mathrm{C}=\frac{1}{2\pi }$$$$\Rightarrow \mathrm{v}\left(\mathrm{t}\right)=\frac{\sin \left(\pi \mathrm{t}\right)}{\pi }+\frac{1}{2\pi }$$$$\mathrm{x}\left(\mathrm{t}\right)=\int \mathrm{v}\left(\mathrm{t}\right)\mathrm{dt}=-\frac{\cos \left(\pi \mathrm{t}\right)}{{\pi }^2}+\frac{\mathrm{t}}{2\pi }+\mathrm{C}$$$$\mathrm{since}\mathrm{x}=0\mathrm{at}\mathrm{t}=0\Rightarrow \mathrm{C}=\frac{1}{{\pi }^2}$$$$\Rightarrow \mathrm{x}\left(\mathrm{t}\right)=\frac{1-\cos \left(\pi \mathrm{t}\right)}{{\pi }^2}+\frac{\mathrm{t}}{2\pi }$$$$$$$$\mathrm{at}\mathrm{t}=\frac{3}{2}$$$$\mathrm{v}=-\frac{1}{\pi }+\frac{1}{2\pi }=-\frac{1}{2\pi }$$$$\mathrm{x}=\frac{1}{{\pi }^2}+\frac{3}{4\pi }$$

Commented by Tinkutara last updated on 07/Jun/17

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

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