8-5 할리데이 11판 솔루션 일반물리학

$$ \begin{cases} F&=72.4x+51.6x^2\\ x_i&=0.500\ut{m}\\ x_f&=1.00\ut{m}\\ m&=1.50\ut{kg} \end{cases} $$ $$\ab{a}$$ $$W_{i\rarr f}=\int_{i}^{f} \vec F \cdot \dd \vec S,$$ $$ \begin{aligned} W_{0.5\rarr1}&=\int_{0.5}^{1} (72.4x+51.6x^2) \cdot \dd x\\ &=\frac{211}{5}\ut{J}\\ &=42.2\ut{J} \end{aligned} $$ $$\ab{b}$$ $$ \put \begin{cases} \KE : \text{Kinetic Energy}\\ \LE : \text{Elastic Potential Energy}\\ \end{cases} $$ $$ \begin{cases} x_1&=1.00\ut{m}\\ x_2&=0.500\ut{m}\\ v_1&=0\\ m&=1.50\ut{kg} \end{cases} $$ $$ \begin{aligned} \Sigma E_1&=\Sigma E_2,\\ \KE_1+\LE_1&=\KE_2+\LE_2\\ \frac{1}{2}m{v_1}^2+\LE_1&=\frac{1}{2}m{v_2}^2+(\LE_1+W_{1\rarr2})\\ \end{aligned} $$ $$0=\frac{1}{2}m{v_2}^2+\int_{x_1}^{x_2} F \cdot \dd x~~(\because v_1=0)$$ $$ \begin{aligned} \frac{1}{2}m{v_2}^2&=\int_{x_2}^{x_1} F \cdot \dd x\\ \end{aligned} $$ $$ \begin{aligned} {v_2}&=\sqrt{\frac{2}{m}\int_{0.5}^{1} F \cdot \dd x}\\ &=2\sqrt\frac{211}{15}\ut{m/s}\\ &\approx 7.501111028818776\ut{m/s}\\ &\approx 7.50\ut{m/s}\\ \end{aligned} $$ $$\ab{c}$$ $$F=\text{Conservative Force}$$