8-40 할리데이 11판 솔루션 일반물리학

$$ \put \begin{cases} \KE : \text{Kinetic Energy}\\ \PE : \text{Potential Energy}\\ \end{cases} $$ $$ \begin{cases} m&=1\ut{kg}\\ F(x)&=-3.0x-5.0x^2\ut{N}\\ \PE_0&=0\\ \end{cases} $$ $$\ab{a}$$ $$ \begin{aligned} \PE(x)&=\PE(0)+\Delta \PE\\ &=\PE(0)+ \int_0^xF\dd x\\ &=0+\int_0^x(-3x-5x^2)\dd x\\ &=-\frac{3}{2}x^2-\frac{5}{3}x^3\\ \end{aligned} $$ $$ \begin{aligned} \PE(2.0\ut{m})&=-\frac{3}{2}(2)^2-\frac{5}{3}(2)^3\\ &=-\frac{58}{3}\ut{J}\\ &\approx -19.333333333333332\ut{J}\\ &\approx -19\ut{J}\\ \end{aligned} $$ $$\ab{b}$$ $$ \begin{cases} \vec v_5&=-4.0\ut{m/s} \end{cases} $$ $$ \begin{aligned} \KE_5&=\frac{1}{2}mv^2\\ &=8\ut{J} \end{aligned} $$ $$ \begin{aligned} \Delta \PE_{5\rarr0} &=\int_5^0(-3x-5x^2)\dd x\\ &=\int_0^5(3x+5x^2)\dd x\\ &=\frac{1475}{6}\ut{J}\\ &\approx 245.83333333333334\ut{J}\\ &\approx 2.5\times 10^2\ut{J}\\ \end{aligned} $$ $$\KE_5 \lt \Delta \PE_{5\rarr0},$$ $$\therefore \text{Impossible to Arrive}$$ $$\ab{c-a}$$ $$ \begin{cases} \PE(0)&=-8.0\ut{J} \end{cases} $$ $$ \begin{aligned} \PE(x)&=\PE(0)+\Delta \PE\\ &=\PE(0)+ \int_0^xF\dd x\\ &=-8+\int_0^x(-3x-5x^2)\dd x\\ &=-8-\frac{3}{2}x^2-\frac{5}{3}x^3\\ \end{aligned} $$ $$ \begin{aligned} \PE(2.0\ut{m})&=-8-\frac{3}{2}(2)^2-\frac{5}{3}(2)^3\\ &=-\frac{82}{3}\ut{J}\\ &\approx -27.333333333333332\ut{J}\\ &\approx -27\ut{J}\\ \end{aligned} $$ $$\ab{c-b}$$ $$ \begin{cases} \vec v_5&=-4.0\ut{m/s} \end{cases} $$ $$ \begin{aligned} \KE_5&=\frac{1}{2}mv^2\\ &=8\ut{J} \end{aligned} $$ $$ \begin{aligned} \Delta \PE_{5\rarr0} &=\int_5^0(-3x-5x^2)\dd x\\ &=\int_0^5(3x+5x^2)\dd x\\ &=\frac{1475}{6}\ut{J}\\ &\approx 245.83333333333334\ut{J}\\ &\approx 2.5\times 10^2\ut{J}\\ \end{aligned} $$ $$\KE_5 \lt \Delta \PE_{5\rarr0},$$ $$\therefore \text{Impossible to Arrive}$$