9-15 할리데이 11판 솔루션 일반물리학

$$ \put \begin{cases} a:m_a\\ b:m_b \end{cases} $$ $$ \begin{cases} v_{ai}&=0.75\ut{m/s}\\ m_a&=0.25\ut{kg}\\ m_b&=0.50\ut{kg}\\ L_a&=5.0\ut{cm}=5\times10^{-2}\ut{m}\\ L_b&=6.0\ut{cm}=6\times10^{-2}\ut{m}\\ x_{ai}&=-1.50\ut{m}\\ x_{bi}&=0\ut{m}\\ \end{cases} $$ $$\put M=m_a+m_b=0.75\ut{kg}$$ $$\vec r_{\com} = \frac{\Sigma m\vec r}{M},$$ $$ \begin{aligned} x_\com&=\frac{\Sigma mx}{M}\\ &=\frac{m_ax_a+m_bx_b}{M} \end{aligned} $$ $$\ab{a}$$ $$t=0,$$ $$ \begin{cases} x_a&=-1.5\ut{m}\\ x_b&=0\\ \end{cases} $$ $$ \begin{aligned} x_\com&=\frac{m_ax_a+m_bx_b}{M}\\ &=\frac{m_a}{M}x_a\\ &=-\frac{1}{2}\ut{m}\\ &=-0.50\ut{m}\\ &=-50\ut{cm}\\ \end{aligned} $$ $$\ab{b}$$ $$t=\text{Contact},$$ $$ \begin{cases} x_a&=-\frac{L_b}{2}-\frac{L_a}{2}=-5.5\ut{cm}\\ x_b&=0\\ \end{cases} $$ $$ \begin{aligned} x_\com&=\frac{m_ax_a+m_bx_b}{M}\\ &=\frac{m_a}{M}x_a\\ &=-\frac{11}{6}\ut{cm}\\ &\approx -1.8333333333333333\ut{cm}\\ &\approx -1.8\ut{cm}\\ \end{aligned} $$ $$\ab{c}$$ $$ \begin{aligned} v_\com&=\dyt{x_\com}\\ &=\dt\frac{m_ax_a+m_bx_b}{M}\\ &=\frac{m_a\dyt{x_a}+m_b\dyt{x_b}}{M}\\ &=\frac{m_av_{ai}+m_bv_{bi}}{M}\\ &=\frac{m_a}{M}v_{ai}\\ \end{aligned} $$ $$ \begin{aligned} x_\com(t)&=x_\com(0)+v_\com t\\ &=\frac{m_a}{M}x_{ai}+\frac{m_a}{M}v_{ai}t\\ &=\frac{m_a}{M}(x_{ai}+v_{ai}t)\\ &=\frac{1}{4}(t-2)\\ \end{aligned} $$ $$ \begin{aligned} x_\com(4)&=\frac{1}{2}\ut{m}\\ &=0.50\ut{m}\\ \end{aligned} $$