4-6 할리데이 11판 솔루션 일반물리학

$$ \begin{cases} &t_1=4.0\ut{min},&v_1=50\ut{km/h}\\ \theta_{1\rarr2}=-90\degree,&t_2=3.0\ut{min},&v_2=20\ut{km/h}\\ \theta_{2\rarr3}=-90\degree,&t_3=60\ut{s},& v_3=20\ut{km/h}\\ \theta_{3\rarr4}=+90\degree,&t_4=60\ut{s},& v_4=50\ut{km/h}\\ \theta_{4\rarr5}=-90\degree,&t_5=2\ut{min},&v_5=20\ut{km/h}\\ \theta_{5\rarr6}=+90\degree,&t_5=30\ut{s},&v_6=50\ut{km/h}\\ \end{cases} $$ $$ \begin{cases} r_1&=4\ut{min}\cdot50\ut{km/h}\cdot\cfrac{1\ut{h}}{60\ut{min}}=\cfrac{10}{3}\ut{km}\\ r_2&=3\ut{min}\cdot20\ut{km/h}\cdot\cfrac{1\ut{h}}{60\ut{min}}=1\ut{km}\\ r_3&=1\ut{min}\cdot20\ut{km/h}\cdot\cfrac{1\ut{h}}{60\ut{min}}=\cfrac{1}{3}\ut{km}\\ r_4&=1\ut{min}\cdot50\ut{km/h}\cdot\cfrac{1\ut{h}}{60\ut{min}}=\cfrac{5}{6}\ut{km}\\ r_5&=2\ut{min}\cdot20\ut{km/h}\cdot\cfrac{1\ut{h}}{60\ut{min}}=\cfrac{2}{3}\ut{km}\\ r_6&=0.5\ut{min}\cdot50\ut{km/h}\cdot\cfrac{1\ut{h}}{60\ut{min}}=\cfrac{5}{12}\ut{km}\\ \end{cases} $$ $$\put \vec r_1=r_1\i,$$ $$ \begin{cases} \vec r_1 &= \cfrac{10}{3}\i\ut{km}\\ \vec r_2 &= -\j\ut{km}\\ \vec r_3 &= -\cfrac{1}{3}\i\ut{km}\\ \vec r_4 &= -\cfrac{5}{6}\j\ut{km}\\ \vec r_5 &= -\cfrac{2}{3}\i\ut{km}\\ \vec r_6 &= -\cfrac{5}{12}\j\ut{km}\\ \end{cases} $$ $$ \begin{aligned} \Sigma \vec r &= \(\cfrac{10}{3}-\cfrac{1}{3}-\cfrac{2}{3}\)\i+\(-1-\cfrac{5}{6}-\cfrac{5}{12}\)\j\\ &=\(\frac{7}{3}\i-\frac{9}{4}\j\)\ut{km}\\ \end{aligned} $$ $$\ab{a}$$ $$ \begin{aligned} r&=\sqrt{\(\frac{7}{3}\)^2+\(-\frac{9}{4}\)^2}\\ &=\frac{\sqrt{1513}}{12}\ut{km}\\ &\approx 3.241441723129454\ut{km}\\ &\approx 3.2\ut{km}\\ \end{aligned} $$ $$\ab{b}$$ $$ \begin{aligned} \theta&=\tan^{-1}\(\frac{-\frac{9}{4}}{\frac{7}{3}}\)\\ &=-\tan ^{-1}\(\frac{27}{28}\)\\ &\approx -0.76721834832447\ut{rad}\\ &\approx -0.77\ut{rad}\\ \end{aligned} $$