9-40 할리데이 11판 솔루션 일반물리학

$$\put \begin{cases} r:\text{red}\\ b:\text{blue} \end{cases} $$ $$ \begin{cases} v_i&=4.00\ut{m/s}\\ \theta&=40.0\degree\\ \end{cases} $$ $$ \begin{cases} v_{ri}&=v_i\cos\theta\i+v_i\sin\theta\j\\ v_{bi}&=v_i\cos\theta\i-v_i\sin\theta\j\\ \end{cases} $$ $$ \Delta p_x = 0, $$ $$ \begin{cases} p_{rix}=p_{rfx}\\ p_{bix}=p_{bfx}\\ \end{cases} $$ $$ \put \begin{cases} v_{\text{in}}&=\vec v_{ri}-\vec v_{bi}=v_{riy}-v_{biy}\\ v_{\text{out}}&=\vec v_{rf}-\vec v_{bf}=v_{rfy}-v_{bfy}\\ \end{cases} $$ $$ \begin{aligned} v_{\text{in}}&=v_{riy}-v_{biy}\\ &=v_{ri}\sin\theta_r-v_{bi}\sin\theta_b\\ &=v_{i}\sin\theta-(-v_{i}\sin\theta)\\ &=2v_{i}\sin\theta\taag1\\ \end{aligned} $$ $$\Delta \Sigma \vec p=0,$$ $$ \begin{aligned} \Sigma \vec p_i&=\Sigma \vec p_f\\ m\vec v_{ri}+m\vec v_{bi}&=m\vec v_{rf}+mv_{bf}\\ \vec v_{ri}+\vec v_{bi}&=\vec v_{rf}+v_{bf}\\ v_{riy}+v_{biy}&=v_{rfy}+v_{bfy}\\ v_{riy}+(-v_{riy})&=v_{rfy}+v_{bfy}\\ 0&=v_{afy}+v_{bfy}\\ \end{aligned} $$ $$ \begin{cases} v_{rfy}&=\cfrac{v_{\text{out}}}{2}\\ v_{bfy}&=-\cfrac{v_{\text{out}}}{2}\\ \end{cases} $$ $$ \begin{cases} \vec v_{rf}&=v_{rix}\i+\cfrac{v_{\text{out}}}{2}\j\\ \vec v_{bf}&=v_{bix}\i-\cfrac{v_{\text{out}}}{2}\j\\ \end{cases} $$ $$\ab{a}$$ $$ \text{Perfectly Inelastic Collision}\Harr\vec v_{\text{out}}=0,$$ $$ \begin{cases} \vec v_{rf}&=v_{rix}\i\\ \vec v_{bf}&=v_{bix}\i\\ \end{cases} $$ $$\text{on Axis }x$$ $$\ab{b}$$ $$ \text{Elastic Collision}\Harr \vec v_{\text{out}}=-\vec v_{\text{in}},$$ $$ \begin{cases} \vec v_{rf}&=v_{rix}\i-\cfrac{v_{\text{in}}}{2}\j\\ \vec v_{bf}&=v_{bix}\i+\cfrac{v_{\text{in}}}{2}\j\\ \end{cases} $$ $$ \begin{cases} \vec v_{rf}&=v_{ri}\cos\theta\i-\cfrac{2v_{i}\sin\theta}{2}\j\\ \vec v_{bf}&=v_{bi}\cos\theta\i+\cfrac{2v_{i}\sin\theta}{2}\j\\ \end{cases} $$ $$ \begin{cases} \vec v_{rf}&=v_{i}\cos\theta\i-v_{i}\sin\theta\j\\ \vec v_{bf}&=v_{i}\cos\theta\i+v_{i}\sin\theta\j\\ \end{cases} $$ $$ \begin{cases} \vec v_{rf}&=\vec v_{bi}\\ \vec v_{bf}&=\vec v_{ri}\\ \end{cases} $$ $$ \begin{cases} \vec v_{rf}&=\text{On Blue Line : 3}\\ \vec v_{bf}&=\text{On Red Line : 2}\\ \end{cases} $$ $$\ab{c}$$ $$ \text{Inelastic Collision}\Harr -v_{\text{in}}\lt v_{\text{out}}\lt 0,$$ $$ -2v_{i}\sin\theta\lt v_{\text{out}}\lt 0\\ -v_{i}\sin\theta\lt \frac{v_{\text{out}}}{2}\lt 0\\ -v_{iy}\lt \frac{v_{\text{out}}}{2}\lt 0\\ -v_{iy}\lt v_f\lt 0\\ $$ $$ \begin{cases} \vec v_{rf}&=v_{rix}\i+\cfrac{v_{\text{out}}}{2}\j\\ \vec v_{bf}&=v_{bix}\i-\cfrac{v_{\text{out}}}{2}\j\\ \end{cases} $$ $$ \begin{cases} \vec v_{rf}&=v_{rix}\i-v_f\j\\ \vec v_{bf}&=v_{bix}\i+v_f\j\\ \end{cases} $$ $$ \begin{cases} \vec v_{rf}&=\text{In Area C}\\ \vec v_{bf}&=\text{In Area B}\\ \end{cases} $$ $$\ab{d}$$ $$ \text{Perfectly Inelastic Collision}\Harr\vec v_{\text{out}}=0,$$ $$ \begin{cases} \vec v_{rf}&=v_{rix}\i\\ \vec v_{bf}&=v_{bix}\i\\ \end{cases} $$ $$ \begin{cases} \vec v_{rf}&=v_{i}\cos\theta\i\\ \vec v_{bf}&=v_{i}\cos\theta\i\\ \end{cases} $$ $$ \begin{aligned} v_f&=v_{rf}=\vec v_{bf}\\ &=v_{i}\cos\theta\\ &=4\cos40\degree\\ &\approx 3.064177772475912\ut{m/s}\\ &\approx 3.06\ut{m/s}\\ \end{aligned} $$ $$\ab{e}$$ $$ \text{Elastic Collision}\Harr \vec v_{\text{out}}=-\vec v_{\text{in}},$$ $$ \begin{cases} \vec v_{rf}&=\vec v_{bi}\\ \vec v_{bf}&=\vec v_{ri}\\ \end{cases} $$ $$ \begin{aligned} v_f&=v_i\\ &=4.00\ut{m/s}\\ \end{aligned} $$