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Question Number 123767 by oustmuchiya@gmail.com last updated on 28/Nov/20

Answered by physicstutes last updated on 28/Nov/20

s = 2t + 3 sin 2t (i) Initiat position occurs at t = 0 s ⇒ s_0 = 2(0) + 3 sin 2(0) s_0 = 0 m (ii) v = (ds/dt) = 2 + 6 cos 2t a = (dv/dt) = 0 − 12 sin 2t ⇒ v =( 2 + 6 cos 2t) m s^(−1) and a = (−12 sin 2t) m s^(−2) (iii) At rest v = 0 ⇒ 2 + 6 cos 2t = 0 cos 2t = −(1/3) 2t = 109.5 ⇒ t = 54.75 s b. F(x) = 250 e^(0.09x) , x ≥ 0 ((dF(x))/dx) = 22.5 e^(0.09x) ((dF(x))/dx)∣_(8 months) = 22.5 e^(0.09×8) = 46.2

$$\:\:\boldsymbol{\mathrm{s}}\:=\:\mathrm{2}{t}\:+\:\mathrm{3}\:\mathrm{sin}\:\mathrm{2}{t} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{Initiat}\:\mathrm{position}\:\mathrm{occurs}\:\mathrm{at}\:{t}\:=\:\mathrm{0}\:\mathrm{s} \\ $$$$\Rightarrow\:\boldsymbol{\mathrm{s}}_{\mathrm{0}} \:=\:\mathrm{2}\left(\mathrm{0}\right)\:+\:\mathrm{3}\:\mathrm{sin}\:\mathrm{2}\left(\mathrm{0}\right) \\ $$$$\:\:\:\:\:\:\boldsymbol{\mathrm{s}}_{\mathrm{0}} \:=\:\mathrm{0}\:\mathrm{m} \\ $$$$\left(\mathrm{ii}\right)\:\boldsymbol{\mathrm{v}}\:=\:\frac{{d}\boldsymbol{\mathrm{s}}}{{dt}}\:=\:\mathrm{2}\:+\:\mathrm{6}\:\mathrm{cos}\:\mathrm{2}{t} \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\mathrm{a}}\:=\:\frac{{d}\boldsymbol{\mathrm{v}}}{{dt}}\:=\:\mathrm{0}\:−\:\mathrm{12}\:\mathrm{sin}\:\mathrm{2}{t} \\ $$$$\Rightarrow\:\boldsymbol{\mathrm{v}}\:=\left(\:\mathrm{2}\:+\:\mathrm{6}\:\mathrm{cos}\:\mathrm{2}{t}\right)\:\mathrm{m}\:\mathrm{s}^{−\mathrm{1}} \:\mathrm{and}\:\boldsymbol{\mathrm{a}}\:=\:\left(−\mathrm{12}\:\mathrm{sin}\:\mathrm{2}{t}\right)\:\mathrm{m}\:\mathrm{s}^{−\mathrm{2}} \\ $$$$\left(\mathrm{iii}\right)\:\mathrm{At}\:\mathrm{rest}\:\boldsymbol{\mathrm{v}}\:=\:\mathrm{0} \\ $$$$\Rightarrow\:\mathrm{2}\:+\:\mathrm{6}\:\mathrm{cos}\:\mathrm{2}{t}\:=\:\mathrm{0} \\ $$$$\:\:\:\mathrm{cos}\:\mathrm{2}{t}\:=\:−\frac{\mathrm{1}}{\mathrm{3}}\: \\ $$$$\:\:\:\:\mathrm{2}{t}\:=\:\mathrm{109}.\mathrm{5}\:\Rightarrow\:{t}\:=\:\mathrm{54}.\mathrm{75}\:\mathrm{s} \\ $$$$\mathrm{b}.\:\mathrm{F}\left({x}\right)\:=\:\mathrm{250}\:{e}^{\mathrm{0}.\mathrm{09}{x}} ,\:{x}\:\geqslant\:\mathrm{0} \\ $$$$\:\:\:\:\frac{{d}\mathrm{F}\left({x}\right)}{{dx}}\:=\:\mathrm{22}.\mathrm{5}\:{e}^{\mathrm{0}.\mathrm{09}{x}} \\ $$$$\:\frac{{d}\mathrm{F}\left({x}\right)}{{dx}}\mid_{\mathrm{8}\:\mathrm{months}} \:=\:\mathrm{22}.\mathrm{5}\:{e}^{\mathrm{0}.\mathrm{09}×\mathrm{8}} \:=\:\mathrm{46}.\mathrm{2}\:\: \\ $$

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