11판/9. 질량중심과 선운동량

9-40 할리데이 11판 솔루션 일반물리학

짱세디럭스 2024. 4. 24. 10:39
put {r:redb:blue\put \begin{cases} r:\text{red}\\ b:\text{blue} \end{cases} {vi=4.00[m/s]θ=40.0° \begin{cases} v_i&=4.00\ut{m/s}\\ \theta&=40.0\degree\\ \end{cases} {vri=vicosθi^+visinθj^vbi=vicosθi^visinθj^ \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} Δpx=0, \Delta p_x = 0, {prix=prfxpbix=pbfx \begin{cases} p_{rix}=p_{rfx}\\ p_{bix}=p_{bfx}\\ \end{cases} put {vin=vrivbi=vriyvbiyvout=vrfvbf=vrfyvbfy \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} vin=vriyvbiy=vrisinθrvbisinθb=visinθ(visinθ)=2visinθ(1) \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} ΔΣp=0,\Delta \Sigma \vec p=0, Σpi=Σpfmvri+mvbi=mvrf+mvbfvri+vbi=vrf+vbfvriy+vbiy=vrfy+vbfyvriy+(vriy)=vrfy+vbfy0=vafy+vbfy \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} {vrfy=vout2vbfy=vout2 \begin{cases} v_{rfy}&=\cfrac{v_{\text{out}}}{2}\\ v_{bfy}&=-\cfrac{v_{\text{out}}}{2}\\ \end{cases} {vrf=vrixi^+vout2j^vbf=vbixi^vout2j^ \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} (a)\ab{a} Perfectly Inelastic Collisionvout=0, \text{Perfectly Inelastic Collision}\Harr\vec v_{\text{out}}=0, {vrf=vrixi^vbf=vbixi^ \begin{cases} \vec v_{rf}&=v_{rix}\i\\ \vec v_{bf}&=v_{bix}\i\\ \end{cases} on Axis x\text{on Axis }x (b)\ab{b} Elastic Collisionvout=vin, \text{Elastic Collision}\Harr \vec v_{\text{out}}=-\vec v_{\text{in}}, {vrf=vrixi^vin2j^vbf=vbixi^+vin2j^ \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} {vrf=vricosθi^2visinθ2j^vbf=vbicosθi^+2visinθ2j^ \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} {vrf=vicosθi^visinθj^vbf=vicosθi^+visinθj^ \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} {vrf=vbivbf=vri \begin{cases} \vec v_{rf}&=\vec v_{bi}\\ \vec v_{bf}&=\vec v_{ri}\\ \end{cases} {vrf=On Blue Line : 3vbf=On Red Line : 2 \begin{cases} \vec v_{rf}&=\text{On Blue Line : 3}\\ \vec v_{bf}&=\text{On Red Line : 2}\\ \end{cases} (c)\ab{c} Inelastic Collisionvin<vout<0, \text{Inelastic Collision}\Harr -v_{\text{in}}\lt v_{\text{out}}\lt 0, 2visinθ<vout<0visinθ<vout2<0viy<vout2<0viy<vf<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\\ {vrf=vrixi^+vout2j^vbf=vbixi^vout2j^ \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} {vrf=vrixi^vfj^vbf=vbixi^+vfj^ \begin{cases} \vec v_{rf}&=v_{rix}\i-v_f\j\\ \vec v_{bf}&=v_{bix}\i+v_f\j\\ \end{cases} {vrf=In Area Cvbf=In Area B \begin{cases} \vec v_{rf}&=\text{In Area C}\\ \vec v_{bf}&=\text{In Area B}\\ \end{cases} (d)\ab{d} Perfectly Inelastic Collisionvout=0, \text{Perfectly Inelastic Collision}\Harr\vec v_{\text{out}}=0, {vrf=vrixi^vbf=vbixi^ \begin{cases} \vec v_{rf}&=v_{rix}\i\\ \vec v_{bf}&=v_{bix}\i\\ \end{cases} {vrf=vicosθi^vbf=vicosθi^ \begin{cases} \vec v_{rf}&=v_{i}\cos\theta\i\\ \vec v_{bf}&=v_{i}\cos\theta\i\\ \end{cases} vf=vrf=vbf=vicosθ=4cos40°3.064177772475912[m/s]3.06[m/s] \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} (e)\ab{e} Elastic Collisionvout=vin, \text{Elastic Collision}\Harr \vec v_{\text{out}}=-\vec v_{\text{in}}, {vrf=vbivbf=vri \begin{cases} \vec v_{rf}&=\vec v_{bi}\\ \vec v_{bf}&=\vec v_{ri}\\ \end{cases} vf=vi=4.00[m/s] \begin{aligned} v_f&=v_i\\ &=4.00\ut{m/s}\\ \end{aligned}