8-54 할리데이 11판 솔루션 일반물리학

$$ \put \begin{cases} \KE : \text{Kinetic Energy}\\ \GE : \text{Gravitational Potential Energy}\\ \LE : \text{Elastic Potential Energy}\\ \end{cases} $$ $$ \begin{cases} k&=620\ut{N/m}\\ x_i&=25\ut{cm}=0.25\ut{m}\\ mg&=50\ut{N}\\ v_i&=0\\ \GE(0)&=0\\ \end{cases} $$ $$ \begin{aligned} \Delta x &= -\Delta h\\ x_f-x_i&=h_i-h_f\\ x_f&=x_i-h_f\\ \end{aligned} $$ $$\Sigma \Delta E=0,$$ $$\Delta \KE+\Delta \GE+\Delta \LE=0$$ $$ \begin{aligned} \KE_f&=\KE_i-\Delta \GE-\frac{1}{2}k\Delta \(x^2\)\\ &=\KE_i-(\GE_f-\GE_i)-\frac{1}{2}k\Delta \(x^2\)\\ &=0-(\GE_f-0)-\frac{1}{2}k\({x_f}^2-{x_i}^2\)\\ &=-mgh_f-\frac{1}{2}k\bra{{\({x_i}-h_f\)}^2-{{x_i}}^2}\\ &=-\frac{1}{2} h_f (2 mg+kh_f -2 k x_i)\\ &=5 h_f (21-62 h_f) \end{aligned} $$ $$\therefore \KE(h)=5 h (21-62 h)$$ $$\ab{a}$$ $$ \KE(0)=0$$ $$\ab{b}$$ $$ \begin{aligned} \KE(0.050\ut{m})&=\frac{179}{40}\ut{J}\\ &=4.475\ut{J}\\ &\approx 4.5\ut{J}\\ \end{aligned} $$ $$\ab{c}$$ $$ \begin{aligned} \KE(0.10\ut{m})&=\frac{37}{5}\ut{J}\\ &=7.4\ut{J}\\ \end{aligned} $$ $$\ab{d}$$ $$ \begin{aligned} \KE(0.15\ut{m})&=\frac{351}{40}\ut{J}\\ &=8.775\ut{J}\\ &\approx 8.8\ut{J}\\ \end{aligned} $$ $$\ab{e}$$ $$ \begin{aligned} \KE(0.2\ut{m})&=\frac{43}{5}\ut{J}\\ &=8.6\ut{J}\\ \end{aligned} $$ $$\ab{f}$$ $$\Sigma \Delta E=0,$$ $$\Delta \KE+\Delta \GE+\Delta \LE=0$$ $$ \begin{aligned} \Delta \GE&=-\Delta \LE\\ \GE_f-\GE_i&=\LE_i-\LE_f\\ \GE_f&=\LE_i\\ mgh&=\frac{1}{2}k{x_i}^2\\ \end{aligned} $$ $$ \begin{aligned} h&=\frac{k{x_i}^2}{2mg}\\ &=\frac{31}{80}\ut{m}\\ &= 0.3875\ut{m}\\ &\approx 0.39\ut{m}\\ \end{aligned} $$