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At-time-t-the-force-acting-on-a-particle-P-of-mass-2kg-is-2ti-4j-N-P-is-initially-at-rest-at-the-point-with-position-vector-i-2j-Find-a-the-velocity-of-P-when-t-2-b-the-position-vector-w

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Question Number 38366 by Rio Mike last updated on 24/Jun/18
At time t,the force acting on a particle P of mass 2kg is (2ti + 4j)N.P is initially at rest at the point with position vector (i + 2j). Find: a) the velocity of P when t = 2. b) the position vector when t = 2.
$${At}\:{time}\:{t},{the}\:{force}\:{acting}\:{on}\:{a}\:{particle} \\ $$$${P}\:{of}\:{mass}\:\mathrm{2}{kg}\:{is}\:\left(\mathrm{2}\boldsymbol{{ti}}\:+\:\mathrm{4}\boldsymbol{{j}}\right){N}.{P} \\ $$$${is}\:{initially}\:{at}\:{rest}\:{at}\:{the}\:{point}\:{with} \\ $$$${position}\:{vector}\:\left(\boldsymbol{{i}}\:+\:\mathrm{2}\boldsymbol{{j}}\right). \\ $$$${Find}: \\ $$$$\left.{a}\right)\:{the}\:{velocity}\:{of}\:{P}\:{when}\:{t}\:=\:\mathrm{2}. \\ $$$$\left.{b}\right)\:{the}\:{position}\:{vector}\:{when}\:{t}\:=\:\mathrm{2}. \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 25/Jun/18
m(dv^→ /dt)=2ti^→ +4j^→ (dv^→ /dt)=ti^→ +2j^→ m=2kg dv^→ =ti^→ dt+2j^→ dt v^→ =(t^2 /2)i^→ +2j^→ t+c t=0 v^→ =0 c=0 v^→ =(t^2 /2)i^→ +2tj^→ v^→ =2i^→ +4j^(→ ) when t=2 (dr^→ /dt)=2i^→ +4j^→ r^ =2ti^→ +4tj^(→r) +c when t=0 r_0 ^→ =i^→ +2j^→ r^→ =2ti^→ +4tj^→ +i^→ +2j^→ r^→ =i^→ (2t+1)+j^→ (4t+2)
$${m}\frac{{d}\overset{\rightarrow} {{v}}}{{dt}}=\mathrm{2}{t}\overset{\rightarrow} {{i}}+\mathrm{4}\overset{\rightarrow} {{j}} \\ $$$$\frac{{d}\overset{\rightarrow} {{v}}}{{dt}}={t}\overset{\rightarrow} {{i}}\:+\mathrm{2}\overset{\rightarrow} {{j}}\:\:\:\:\:{m}=\mathrm{2}{kg} \\ $$$${d}\overset{\rightarrow} {{v}}={t}\overset{\rightarrow} {{i}dt}+\mathrm{2}\overset{\rightarrow} {{j}dt} \\ $$$$\overset{\rightarrow} {{v}}=\frac{{t}^{\mathrm{2}} }{\mathrm{2}}\overset{\rightarrow} {{i}}+\mathrm{2}\overset{\rightarrow} {{j}t}+{c} \\ $$$${t}=\mathrm{0}\:\:\overset{\rightarrow} {{v}}=\mathrm{0}\:\:\:{c}=\mathrm{0} \\ $$$$\overset{\rightarrow} {{v}}=\frac{{t}^{\mathrm{2}} }{\mathrm{2}}\overset{\rightarrow} {{i}}+\mathrm{2}{t}\overset{\rightarrow} {{j}} \\ $$$$\overset{\rightarrow} {{v}}=\mathrm{2}\overset{\rightarrow} {{i}}+\mathrm{4}\overset{\rightarrow\:} {{j}}\:\:\:\:{when}\:{t}=\mathrm{2} \\ $$$$\frac{{d}\overset{\rightarrow} {{r}}}{{dt}}=\mathrm{2}\overset{\rightarrow} {{i}}+\mathrm{4}\overset{\rightarrow} {{j}} \\ $$$$\overset{} {{r}}=\mathrm{2}{t}\overset{\rightarrow} {{i}}\:+\mathrm{4}{t}\overset{\rightarrow{r}} {{j}}+{c} \\ $$$${when}\:{t}=\mathrm{0}\:\:\:\overset{\rightarrow} {{r}}_{\mathrm{0}} =\overset{\rightarrow} {{i}}+\mathrm{2}\overset{\rightarrow} {{j}} \\ $$$$\overset{\rightarrow} {{r}}=\mathrm{2}{t}\overset{\rightarrow} {{i}}+\mathrm{4}{t}\overset{\rightarrow} {{j}}+\overset{\rightarrow} {{i}}+\mathrm{2}\overset{\rightarrow} {{j}} \\ $$$$\overset{\rightarrow} {{r}}=\overset{\rightarrow} {{i}}\left(\mathrm{2}{t}+\mathrm{1}\right)+\overset{\rightarrow} {{j}}\left(\mathrm{4}{t}+\mathrm{2}\right) \\ $$

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