Question Number 38365 by Rio Mike last updated on 24/Jun/18
$${A}\:{particle}\:{P}\:{moves}\:{on}\:{a}\:{straightline} \\ $$$${from}\:{a}\:{fixed}\:{point}\:{O}\:{and}\:{the}\:{distance} \\ $$$${x}\:{from}\:{O}\:{after}\:{t}\:{seconds}\:{is}\:{given}\:{as} \\ $$$$\:{x}\:=\:\frac{\mathrm{1}}{\mathrm{4}\:}\:{t}^{\mathrm{4}} \:−\:\frac{\mathrm{3}}{\mathrm{2}}\:{t}^{\mathrm{2}} \:+\:\mathrm{2}{t}. \\ $$$${Find}: \\ $$$$\left.{a}\right)\:{the}\:{velocity}\:{of}\:{P}\:{when}\:{t}\:=\:\mathrm{2}, \\ $$$$\left.{b}\right)\:{the}\:{acceleration}\:{of}\:{P}\:{when}\:{t}\:=\:\mathrm{2}, \\ $$$$\left.{c}\right)\:{the}\:{time}\:{at}\:{which}\:{the}\:{speed}\:{P}\: \\ $$$${is}\:{Minimum}. \\ $$
Answered by MJS last updated on 25/Jun/18
$${s}\left({t}\right)=\frac{\mathrm{1}}{\mathrm{4}}{t}^{\mathrm{4}} −\frac{\mathrm{3}}{\mathrm{2}}{t}^{\mathrm{2}} +\mathrm{2}{t} \\ $$$${v}\left({t}\right)=\frac{{ds}}{{dt}}={t}^{\mathrm{3}} −\mathrm{3}{t}+\mathrm{2}\:\Rightarrow\:{v}\left(\mathrm{2}\right)=\mathrm{4} \\ $$$${a}\left({t}\right)=\frac{{dv}}{{dt}}=\mathrm{3}{t}^{\mathrm{2}} −\mathrm{3}\:\Rightarrow\:{a}\left(\mathrm{2}\right)=\mathrm{9} \\ $$$$\mathrm{local}\:\mathrm{min}\left({v}\left({t}\right)\right)/\mathrm{max}\left({v}\left({t}\right)\right)={v}\left({x}\right)\:\Rightarrow\:{v}'\left({x}\right)=\mathrm{0}\:\Rightarrow\:{a}\left({x}\right)=\mathrm{0} \\ $$$${v}\left({x}\right)=\mathrm{min}\left({v}\left({t}\right)\right)\:\Rightarrow\:{v}''\left({x}\right)>\mathrm{0} \\ $$$${v}\left({x}\right)=\mathrm{max}\left({v}\left({t}\right)\right)\:\Rightarrow\:{v}''\left({x}\right)<\mathrm{0} \\ $$$$\mathrm{3}{x}^{\mathrm{2}} −\mathrm{3}=\mathrm{0} \\ $$$${x}^{\mathrm{2}} =\mathrm{1} \\ $$$${x}_{\mathrm{1}} =−\mathrm{1} \\ $$$${x}_{\mathrm{2}} =\mathrm{1} \\ $$$${v}''\left({t}\right)={a}'\left({t}\right)=\frac{{da}}{{dt}}=\mathrm{3}{t} \\ $$$${a}'\left(−\mathrm{1}\right)=−\mathrm{3}\:<\mathrm{0}\:\Rightarrow\:\mathrm{local}\:\mathrm{max}\left({v}\left({t}\right)\right)={v}\left(−\mathrm{1}\right)=\mathrm{4} \\ $$$${a}'\left(\mathrm{1}\right)=\mathrm{3}\:>\mathrm{0}\:\Rightarrow\:\mathrm{local}\:\mathrm{min}\left({v}\left({t}\right)\right)={v}\left(\mathrm{1}\right)=\mathrm{0} \\ $$
Commented by Rio Mike last updated on 25/Jun/18
$${perfect}! \\ $$