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Question Number 184832 by Mastermind last updated on 12/Jan/23

Given the acceleration a=−4sin2t, initial velocity v(0)=2, and the initial position of the body as s(0)=−3, find the body′s position at time t. Hi

$$\mathrm{Given}\:\mathrm{the}\:\mathrm{acceleration}\: \\ $$$$\mathrm{a}=−\mathrm{4sin2t},\:\mathrm{initial}\:\mathrm{velocity}\: \\ $$$$\mathrm{v}\left(\mathrm{0}\right)=\mathrm{2},\:\mathrm{and}\:\mathrm{the}\:\mathrm{initial}\:\mathrm{position}\: \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{body}\:\mathrm{as}\:\mathrm{s}\left(\mathrm{0}\right)=−\mathrm{3},\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{body}'\mathrm{s}\:\mathrm{position}\:\mathrm{at}\:\mathrm{time}\:\mathrm{t}. \\ $$$$ \\ $$$$\mathrm{Hi} \\ $$

Commented by manolex last updated on 12/Jan/23

don′t believe, CHECK you alcohol s=4sin2t−3 s=sin2t−3 s(0)=−3 v=8cos2t v=2cos2t v(0)=−2 a=−16sin2t a=−4sin2t wrong ok

$${don}'{t}\:{believe},\:\:{CHECK} \\ $$$${you}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{alcohol} \\ $$$${s}=\mathrm{4}{sin}\mathrm{2}{t}−\mathrm{3}\:\:\:\:\:{s}={sin}\mathrm{2}{t}−\mathrm{3} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{s}\left(\mathrm{0}\right)=−\mathrm{3} \\ $$$${v}=\mathrm{8}{cos}\mathrm{2}{t}\:\:\:\:\:\:\:\:\:\:\:\:{v}=\mathrm{2}{cos}\mathrm{2}{t} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{v}\left(\mathrm{0}\right)=−\mathrm{2} \\ $$$${a}=−\mathrm{16}{sin}\mathrm{2}{t}\:\:\:\:\:{a}=−\mathrm{4}{sin}\mathrm{2}{t} \\ $$$${wrong}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{ok} \\ $$

Answered by TheSupreme last updated on 12/Jan/23

v(t)=v(0)+∫_0 ^t adt=2+2cos(2t) s(t)=s(0)+∫vdt=−3+2t+sin(2t)

$${v}\left({t}\right)={v}\left(\mathrm{0}\right)+\int_{\mathrm{0}} ^{{t}} {adt}=\mathrm{2}+\mathrm{2}{cos}\left(\mathrm{2}{t}\right) \\ $$$${s}\left({t}\right)={s}\left(\mathrm{0}\right)+\int{vdt}=−\mathrm{3}+\mathrm{2}{t}+{sin}\left(\mathrm{2}{t}\right) \\ $$$$ \\ $$

Commented by Mastermind last updated on 12/Jan/23

I got s=4sin2t−3 Anyways, thank you

$$\mathrm{I}\:\mathrm{got}\:\mathrm{s}=\mathrm{4sin2t}−\mathrm{3} \\ $$$$ \\ $$$$\mathrm{Anyways},\:\mathrm{thank}\:\mathrm{you} \\ $$

Commented by JDamian last updated on 12/Jan/23

which is obviously wrong. Have you tried to get s'' and compare with a?

Answered by alcohol last updated on 12/Jan/23

a = (dv/(dt )) ⇒ v = 2cos(2t) + k k = 2 − 2cos(0) = 0 ⇒ k = 0 ⇒ v = 2cos(2t) v = (ds/dt) ⇒ s = sin(2t) + c c = −3 − sin(0) = −3 ⇒ s = sin(2t) − 3 Alcohol

$$ \\ $$$${a}\:=\:\frac{{dv}}{{dt}\:}\:\Rightarrow\:{v}\:=\:\mathrm{2}{cos}\left(\mathrm{2}{t}\right)\:+\:{k} \\ $$$${k}\:=\:\mathrm{2}\:−\:\mathrm{2}{cos}\left(\mathrm{0}\right)\:=\:\mathrm{0} \\ $$$$\Rightarrow\:{k}\:=\:\mathrm{0}\:\Rightarrow\:{v}\:=\:\mathrm{2}{cos}\left(\mathrm{2}{t}\right) \\ $$$${v}\:=\:\frac{{ds}}{{dt}}\:\Rightarrow\:{s}\:=\:{sin}\left(\mathrm{2}{t}\right)\:+\:{c} \\ $$$${c}\:=\:−\mathrm{3}\:−\:{sin}\left(\mathrm{0}\right)\:=\:−\mathrm{3} \\ $$$$\Rightarrow\:{s}\:=\:{sin}\left(\mathrm{2}{t}\right)\:−\:\mathrm{3} \\ $$$$ \\ $$$$\mathrm{Alcohol} \\ $$

Answered by Frix last updated on 12/Jan/23

s(t)=∫(∫a(t)dt)dt ∫(∫(−4sin 2t)dt)dt=∫(2cos 2t +C_1 )dt= =sin 2t +C_1 t+C_2 v(0)=2 2cos 0 +C_1 =2 ⇒ C_1 =0 s(0)=−3 sin 0 +C_2 =−3 ⇒ C_2 =−3 ⇒ s(t)=−3+sin 2t Apple Juice

$${s}\left({t}\right)=\int\left(\int{a}\left({t}\right){dt}\right){dt} \\ $$$$\int\left(\int\left(−\mathrm{4sin}\:\mathrm{2}{t}\right){dt}\right){dt}=\int\left(\mathrm{2cos}\:\mathrm{2}{t}\:+{C}_{\mathrm{1}} \right){dt}= \\ $$$$=\mathrm{sin}\:\mathrm{2}{t}\:+{C}_{\mathrm{1}} {t}+{C}_{\mathrm{2}} \\ $$$${v}\left(\mathrm{0}\right)=\mathrm{2} \\ $$$$\mathrm{2cos}\:\mathrm{0}\:+{C}_{\mathrm{1}} =\mathrm{2}\:\Rightarrow\:{C}_{\mathrm{1}} =\mathrm{0} \\ $$$${s}\left(\mathrm{0}\right)=−\mathrm{3} \\ $$$$\mathrm{sin}\:\mathrm{0}\:+{C}_{\mathrm{2}} =−\mathrm{3}\:\Rightarrow\:{C}_{\mathrm{2}} =−\mathrm{3} \\ $$$$\Rightarrow \\ $$$${s}\left({t}\right)=−\mathrm{3}+\mathrm{sin}\:\mathrm{2}{t} \\ $$$$ \\ $$$$\mathrm{Apple}\:\mathrm{Juice} \\ $$

Commented by Rasheed.Sindhi last updated on 12/Jan/23

Appreciate your sense of humour!

$${Appreciate}\:{your}\:{sense}\:{of}\:{humour}! \\ $$

Commented by Frix last updated on 12/Jan/23

It′s better to laugh than to cry!

$$\mathrm{It}'\mathrm{s}\:\mathrm{better}\:\mathrm{to}\:\mathrm{laugh}\:\mathrm{than}\:\mathrm{to}\:\mathrm{cry}! \\ $$

Commented by Rasheed.Sindhi last updated on 12/Jan/23

Well said sir!

Answered by a.lgnaoui last updated on 13/Jan/23

v=∫_0 ^t adt=[2cos 2t+c]_0 ^t =2sin 2t +v_0 [c=v_0 ] v=2cos 2t+2 S=∫_0 ^t vdt=sin 2t+2t+s_0 t=0 s_0 =−3 x=sin 2t+2t−3

$${v}=\int_{\mathrm{0}} ^{{t}} {adt}=\left[\mathrm{2cos}\:\:\mathrm{2}{t}+{c}\right]_{\mathrm{0}} ^{{t}} =\mathrm{2sin}\:\mathrm{2}{t}\:+{v}_{\mathrm{0}} \:\:\left[{c}={v}_{\mathrm{0}} \right] \\ $$$${v}=\mathrm{2cos}\:\:\mathrm{2}{t}+\mathrm{2} \\ $$$${S}=\int_{\mathrm{0}} ^{{t}} {vdt}=\mathrm{sin}\:\mathrm{2}{t}+\mathrm{2}{t}+{s}_{\mathrm{0}} \\ $$$${t}=\mathrm{0}\:\:\:\:\:{s}_{\mathrm{0}} =−\mathrm{3} \\ $$$$ \\ $$$$\:\:\:\:{x}=\mathrm{sin}\:\:\mathrm{2}{t}+\mathrm{2}{t}−\mathrm{3} \\ $$$$ \\ $$

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