Question and Answers Forum

All Questions      Topic List

Mechanics Questions

Previous in All Question      Next in All Question      

Previous in Mechanics      Next in Mechanics      

Question Number 63073 by ajfour last updated on 28/Jun/19

Commented by ajfour last updated on 28/Jun/19

Find x_(min)  .

$${Find}\:{x}_{{min}} \:. \\ $$

Answered by mr W last updated on 28/Jun/19

Commented by mr W last updated on 28/Jun/19

absolute min. s=x_0 =20m.  min. s is at v=0, point B.  min. s ≈ 84 m

$${absolute}\:{min}.\:{s}={x}_{\mathrm{0}} =\mathrm{20}{m}. \\ $$$${min}.\:{s}\:{is}\:{at}\:{v}=\mathrm{0},\:{point}\:{B}. \\ $$$${min}.\:{s}\:\approx\:\mathrm{84}\:{m} \\ $$

Commented by mr W last updated on 28/Jun/19

Commented by mr W last updated on 28/Jun/19

to be exact:  Δt=(2/(36))×4=(2/9) sec  Δs=−2×(2/9)+(1/2)×9×((2/9))^2 =−(2/9) m  ⇒min. s (at point B)=84−(2/9)≈83.78 m  at t=5(2/9) sec

$${to}\:{be}\:{exact}: \\ $$$$\Delta{t}=\frac{\mathrm{2}}{\mathrm{36}}×\mathrm{4}=\frac{\mathrm{2}}{\mathrm{9}}\:{sec} \\ $$$$\Delta{s}=−\mathrm{2}×\frac{\mathrm{2}}{\mathrm{9}}+\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{9}×\left(\frac{\mathrm{2}}{\mathrm{9}}\right)^{\mathrm{2}} =−\frac{\mathrm{2}}{\mathrm{9}}\:{m} \\ $$$$\Rightarrow{min}.\:{s}\:\left({at}\:{point}\:{B}\right)=\mathrm{84}−\frac{\mathrm{2}}{\mathrm{9}}\approx\mathrm{83}.\mathrm{78}\:{m} \\ $$$${at}\:{t}=\mathrm{5}\frac{\mathrm{2}}{\mathrm{9}}\:{sec} \\ $$

Answered by mr W last updated on 28/Jun/19

analytical solution:  v(t)=12+5t for 0≤t≤2  v(2)=12+10=22  v(t)=22−(t−2)8=38−8t for 2≤t≤5  v(5)=38−40=−2  v(t)=−2+9(t−5)=−47+9t for 5≤t≤9  v=0⇒−47+9t=0 ⇒t=((47)/9)  s(t)=20+12t+(1/2)5t^2  for 0≤t≤2  s(2)=20+24+10=54  s(t)=54+22(t−2)−(8/2)(t−2)^2  for 2≤t≤5  s(5)=54+22×3−4×3^2 =84  s(t)=84−2(t−5)+(9/2)(t−5)^2  for 5≤t≤9  min. s=s(((47)/9))=84−2×(((47)/9)−5)+(9/2)(((47)/9)−5)^2 =83.78 m

$${analytical}\:{solution}: \\ $$$${v}\left({t}\right)=\mathrm{12}+\mathrm{5}{t}\:{for}\:\mathrm{0}\leqslant{t}\leqslant\mathrm{2} \\ $$$${v}\left(\mathrm{2}\right)=\mathrm{12}+\mathrm{10}=\mathrm{22} \\ $$$${v}\left({t}\right)=\mathrm{22}−\left({t}−\mathrm{2}\right)\mathrm{8}=\mathrm{38}−\mathrm{8}{t}\:{for}\:\mathrm{2}\leqslant{t}\leqslant\mathrm{5} \\ $$$${v}\left(\mathrm{5}\right)=\mathrm{38}−\mathrm{40}=−\mathrm{2} \\ $$$${v}\left({t}\right)=−\mathrm{2}+\mathrm{9}\left({t}−\mathrm{5}\right)=−\mathrm{47}+\mathrm{9}{t}\:{for}\:\mathrm{5}\leqslant{t}\leqslant\mathrm{9} \\ $$$${v}=\mathrm{0}\Rightarrow−\mathrm{47}+\mathrm{9}{t}=\mathrm{0}\:\Rightarrow{t}=\frac{\mathrm{47}}{\mathrm{9}} \\ $$$${s}\left({t}\right)=\mathrm{20}+\mathrm{12}{t}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{5}{t}^{\mathrm{2}} \:{for}\:\mathrm{0}\leqslant{t}\leqslant\mathrm{2} \\ $$$${s}\left(\mathrm{2}\right)=\mathrm{20}+\mathrm{24}+\mathrm{10}=\mathrm{54} \\ $$$${s}\left({t}\right)=\mathrm{54}+\mathrm{22}\left({t}−\mathrm{2}\right)−\frac{\mathrm{8}}{\mathrm{2}}\left({t}−\mathrm{2}\right)^{\mathrm{2}} \:{for}\:\mathrm{2}\leqslant{t}\leqslant\mathrm{5} \\ $$$${s}\left(\mathrm{5}\right)=\mathrm{54}+\mathrm{22}×\mathrm{3}−\mathrm{4}×\mathrm{3}^{\mathrm{2}} =\mathrm{84} \\ $$$${s}\left({t}\right)=\mathrm{84}−\mathrm{2}\left({t}−\mathrm{5}\right)+\frac{\mathrm{9}}{\mathrm{2}}\left({t}−\mathrm{5}\right)^{\mathrm{2}} \:{for}\:\mathrm{5}\leqslant{t}\leqslant\mathrm{9} \\ $$$${min}.\:{s}={s}\left(\frac{\mathrm{47}}{\mathrm{9}}\right)=\mathrm{84}−\mathrm{2}×\left(\frac{\mathrm{47}}{\mathrm{9}}−\mathrm{5}\right)+\frac{\mathrm{9}}{\mathrm{2}}\left(\frac{\mathrm{47}}{\mathrm{9}}−\mathrm{5}\right)^{\mathrm{2}} =\mathrm{83}.\mathrm{78}\:{m} \\ $$

Commented by mr W last updated on 28/Jun/19

Commented by ajfour last updated on 29/Jun/19

Very elegant Sir, (cant surmise  how you plotted?)!

$${Very}\:{elegant}\:{Sir},\:\left({cant}\:{surmise}\right. \\ $$$$\left.{how}\:{you}\:{plotted}?\right)! \\ $$

Commented by mr W last updated on 29/Jun/19

i use the app Grapher for plotting  the curves. actually there are three  curves. but for each curve you can  set its range. this makes them look  like a single curve. try it! it′s a good  app.

$${i}\:{use}\:{the}\:{app}\:{Grapher}\:{for}\:{plotting} \\ $$$${the}\:{curves}.\:{actually}\:{there}\:{are}\:{three} \\ $$$${curves}.\:{but}\:{for}\:{each}\:{curve}\:{you}\:{can} \\ $$$${set}\:{its}\:{range}.\:{this}\:{makes}\:{them}\:{look} \\ $$$${like}\:{a}\:{single}\:{curve}.\:{try}\:{it}!\:{it}'{s}\:{a}\:{good} \\ $$$${app}. \\ $$

Commented by mr W last updated on 29/Jun/19

Terms of Service

Privacy Policy

Contact: info@tinkutara.com