2-52 할리데이 10판 솔루션 일반물리학

$$\begin{cases} t_{1} = \text{Last 20\% Point}\\ t_{2} = \text{Landing}\\ \end{cases}$$ $$\begin{cases} x_0 = 90\ut{m}\\ x_1 = \frac{20}{100}x_0=\frac{1}{5}x_0\\ x_2 = 0\\ v_0 = 0\\ a = -g = -9.80665\ut{m/s^2}\\ \end{cases}$$
(a)$t_{1\to2}=?$ $$\Delta x = v_0t+\frac{1}{2}at^2,$$ $$\Delta x_{0\to t} = \frac{1}{2}(-g)t^2$$ $$\therefore t=\sqrt{-\frac{2(x_t-x_0)}{g} }$$ $$ \begin{aligned} t_1&=\sqrt{-\frac{2(x_1-x_0)}{g} }\\ &=\sqrt{-\frac{2(\frac{20}{100}x_0-x_0)}{g} }\\ &=\sqrt{\frac{8x_0}{5g} }\\ \end{aligned} $$ $$ \begin{aligned} t_2&=\sqrt{-\frac{2(x_2-x_0)}{g} }\\ &=\sqrt{\frac{2x_0}{g} } \end{aligned} $$ $$ \begin{aligned} \Ans &= t_2-t_1\\ &= \sqrt{\frac{2x_0}{g} }-\sqrt{\frac{8x_0}{5g} }\\ &=\(\sqrt{2}-\sqrt{\frac{8}{5}}\)\sqrt{\frac{x_0}{g}}\\ &=\(\sqrt{2}-2\sqrt{\frac{2}{5}}\)\sqrt{ \frac{90\ut{m}}{9.80665\ut{m/s^2}}}\\ &= \left(\sqrt{2}-2\sqrt{\frac{2}{5}}\right)600 \sqrt{\frac{5}{196133}} \ut{s}\\ &\approx 0.4523014623143989\ut{s}\\ &\approx 0.45\ut{s} \end{aligned} $$
(b)$v_1=?$ $$ 2a\Delta x = v^2-v_0^2,$$ $$ \begin{aligned} v_1^2&= 2(-g)\Delta x_{0\to1} +v_0^2\\ v_1&=\sqrt{ -2g(x_1-x_0)}\\ &=\sqrt{ -2g\(\frac{1}{5}x_0-x_0\)}\\ &=\sqrt{ \frac{8}{5}gx_0}\\ &=\sqrt{ \frac{8}{5}(9.80665\ut{m/s^2})(90\ut{m})}\\ &=\frac{3 }{25}\sqrt{\frac{196133}{2}}\ut{m/s}\\ &\approx 37.57868544800363\ut{m/s}\\ &\approx 38\ut{m/s} \end{aligned} $$
(c)$v_2=?$ $$ 2a\Delta x = v^2-v_0^2,$$ $$ \begin{aligned} v_2^2&= 2(-g)\Delta x_{0\to2} +v_0^2\\ v_2&=\sqrt{ -2g(x_2-x_0)}\\ &=\sqrt{ 2gx_0}\\ &=\sqrt{ 2(9.80665\ut{m/s^2})(90\ut{m})}\\ &=\frac{3}{10}\sqrt{\frac{196133}{2}}\ut{m/s}\\ &\approx 42.01424758340913\ut{m/s}\\ &\approx 42\ut{m/s}\\ \end{aligned} $$