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Question Number 13578 by tawa tawa last updated on 21/May/17
Answered by ajfour last updated on 21/May/17
Commented by tawa tawa last updated on 21/May/17
Gid bless you sir. Any calculation ?
$$\mathrm{Gid}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}.\:\mathrm{Any}\:\mathrm{calculation}\:? \\ $$
Answered by ajfour last updated on 21/May/17
Answered by ajfour last updated on 21/May/17
a= { ((18 ; 0<t<20)),((−5 ; 20<t<92)) :} v= { ((18t ; 0≤t≤20)),((5(92−t) ; 20<t≤92)) :} x= { ((9t^2 ; 0≤t≤20)),((16560−(5/2)(t−92)^2 ; 20≤t≤92)) :}
$$\mathrm{a}=\begin{cases}{\mathrm{18}\:\:;\:\mathrm{0}<\mathrm{t}<\mathrm{20}}\\{−\mathrm{5}\:\:;\:\mathrm{20}<\mathrm{t}<\mathrm{92}}\end{cases} \\ $$$$\mathrm{v}=\begin{cases}{\mathrm{18t}\:\:;\:\mathrm{0}\leqslant\mathrm{t}\leqslant\mathrm{20}}\\{\mathrm{5}\left(\mathrm{92}−\mathrm{t}\right)\:\:;\:\mathrm{20}<\mathrm{t}\leqslant\mathrm{92}}\end{cases} \\ $$$$\mathrm{x}=\begin{cases}{\mathrm{9t}^{\mathrm{2}} \:\:;\:\:\mathrm{0}\leqslant\mathrm{t}\leqslant\mathrm{20}}\\{\mathrm{16560}−\frac{\mathrm{5}}{\mathrm{2}}\left(\mathrm{t}−\mathrm{92}\right)^{\mathrm{2}} \:\:;\:\mathrm{20}\leqslant\mathrm{t}\leqslant\mathrm{92}}\end{cases} \\ $$
Answered by mrW1 last updated on 21/May/17
a(t)=18, for 0≤t≤20 a(t)=−5, for t≥20 v(t)=∫adt=∫18dt=18t+C_1 for t≤20 v(t)=∫adt=−∫5dt=−5t+C_2 for t≥20 v=0 at t=0 ⇒C_1 =0 ⇒v(t)=18t, for t≤20 at t=20, v=18×20=360 m/s 360=−5×20+C_2 ⇒C_2 =460 ⇒v(t)=−5t+460, for t≥20 time the car stops: 0=−5t+460 t=((460)/5)=92 s x(t)=∫vdt=∫18tdt=9t^2 +C_3 , for t≤20 x(t)=∫vdt=∫(−5t+460)dt=−(5/2)t^2 +460t+C_4 , for t≥20 at t=0, x=0⇒C_3 =0 ⇒x(t)=9t^2 , for t≤20 at t=20: x=9×20^2 =3600 m 3600=−(5/2)×20^2 +460×20+C_4 ⇒C_4 =−4600 ⇒x(t)=−(5/2)t^2 +460t−4600, for t≥20 at t=92: x=−(5/2)×92^2 +460×92−4600=16560 m
$${a}\left({t}\right)=\mathrm{18},\:{for}\:\mathrm{0}\leqslant{t}\leqslant\mathrm{20} \\ $$$${a}\left({t}\right)=−\mathrm{5},\:{for}\:{t}\geqslant\mathrm{20} \\ $$$$ \\ $$$${v}\left({t}\right)=\int{adt}=\int\mathrm{18}{dt}=\mathrm{18}{t}+{C}_{\mathrm{1}} \:{for}\:{t}\leqslant\mathrm{20} \\ $$$${v}\left({t}\right)=\int{adt}=−\int\mathrm{5}{dt}=−\mathrm{5}{t}+{C}_{\mathrm{2}} \:{for}\:{t}\geqslant\mathrm{20} \\ $$$$ \\ $$$${v}=\mathrm{0}\:{at}\:{t}=\mathrm{0}\:\Rightarrow{C}_{\mathrm{1}} =\mathrm{0} \\ $$$$\Rightarrow{v}\left({t}\right)=\mathrm{18}{t},\:{for}\:{t}\leqslant\mathrm{20} \\ $$$${at}\:{t}=\mathrm{20},\:{v}=\mathrm{18}×\mathrm{20}=\mathrm{360}\:{m}/{s} \\ $$$$\mathrm{360}=−\mathrm{5}×\mathrm{20}+{C}_{\mathrm{2}} \\ $$$$\Rightarrow{C}_{\mathrm{2}} =\mathrm{460} \\ $$$$\Rightarrow{v}\left({t}\right)=−\mathrm{5}{t}+\mathrm{460},\:{for}\:{t}\geqslant\mathrm{20} \\ $$$$ \\ $$$${time}\:{the}\:{car}\:{stops}: \\ $$$$\mathrm{0}=−\mathrm{5}{t}+\mathrm{460} \\ $$$${t}=\frac{\mathrm{460}}{\mathrm{5}}=\mathrm{92}\:{s} \\ $$$$ \\ $$$${x}\left({t}\right)=\int{vdt}=\int\mathrm{18}{tdt}=\mathrm{9}{t}^{\mathrm{2}} +{C}_{\mathrm{3}} ,\:{for}\:{t}\leqslant\mathrm{20} \\ $$$${x}\left({t}\right)=\int{vdt}=\int\left(−\mathrm{5}{t}+\mathrm{460}\right){dt}=−\frac{\mathrm{5}}{\mathrm{2}}{t}^{\mathrm{2}} +\mathrm{460}{t}+{C}_{\mathrm{4}} ,\:{for}\:{t}\geqslant\mathrm{20} \\ $$$${at}\:{t}=\mathrm{0},\:{x}=\mathrm{0}\Rightarrow{C}_{\mathrm{3}} =\mathrm{0} \\ $$$$\Rightarrow{x}\left({t}\right)=\mathrm{9}{t}^{\mathrm{2}} ,\:{for}\:{t}\leqslant\mathrm{20} \\ $$$${at}\:{t}=\mathrm{20}:\:{x}=\mathrm{9}×\mathrm{20}^{\mathrm{2}} =\mathrm{3600}\:{m} \\ $$$$\mathrm{3600}=−\frac{\mathrm{5}}{\mathrm{2}}×\mathrm{20}^{\mathrm{2}} +\mathrm{460}×\mathrm{20}+{C}_{\mathrm{4}} \\ $$$$\Rightarrow{C}_{\mathrm{4}} =−\mathrm{4600} \\ $$$$\Rightarrow{x}\left({t}\right)=−\frac{\mathrm{5}}{\mathrm{2}}{t}^{\mathrm{2}} +\mathrm{460}{t}−\mathrm{4600},\:{for}\:{t}\geqslant\mathrm{20} \\ $$$$ \\ $$$${at}\:{t}=\mathrm{92}: \\ $$$${x}=−\frac{\mathrm{5}}{\mathrm{2}}×\mathrm{92}^{\mathrm{2}} +\mathrm{460}×\mathrm{92}−\mathrm{4600}=\mathrm{16560}\:{m} \\ $$

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