2-38 할리데이 10판 솔루션 일반물리학

$$\begin{cases} |a| &= 1.34\ut{m/s^2} \\ x_0 &= 0 \\ x_2 &= 806\ut{m} \\ v_0 &= 0 \\ v_2 &= 0 \end{cases}$$
(a) $\max v = v_M = v_1?$ $$2a\Delta x = v^2-v_0^2,$$ $$\text{(Make System of Equations)}$$ $$\begin{cases} 2a_{0 \to 1}\Delta x_{0 \to 1} &= v_1^2-v_0^2 \\ 2a_{1 \to 2}\Delta x_{1 \to 2} &= v_2^2-v_1^2 \end{cases}$$ $$\begin{cases} 2a_{0 \to 1}\underset{x}{x_1} &= \underset{y}{v_1}^2 \\ 2(-a_{0 \to 1})(x_2-\underset{x}{x_1}) &= -\underset{y}{v_1}^2 \end{cases}$$ $$\text{(UnderSet x,y Means Unknown Value)}$$ $$x_1=\frac{x_2}{2} = \frac{806}{2} = 403\ut{m}$$ $$\begin{aligned} v_1&=\pm \sqrt{a_{0 \to 1}x_2} \\ &= \sqrt{(1.34\ut{m/s^2})(806\ut{m})}\ (\because v_1 > 0) \\ &=\frac{\sqrt{27001}}{5}\ut{m/s} \\ &\approx 32.86396202529452\ut{m/s} \\ &\approx 32.9\ut{m/s} \end{aligned}$$
(b) $t_{0 \to 2} =?$ $$v=v_0+at,$$ $$\begin{aligned} t&=\frac{v-v_0}{a} \\t_{0 \to 2}&=t_{0 \to 1}+t_{1 \to 2} \\&=\frac{v_1-v_0}{a_{0 \to 1}}+\frac{v_2-v_1}{a_{1 \to 2}} \\&=\frac{v_1}{a_{0 \to 1}}+\frac{-v_1}{-a_{0 \to 1}} \\&=2\frac{v_1}{a_{0 \to 1}} \, (=2t_{0 \to 1}) \\&=2\frac{\frac{\sqrt{27001}}{5}\ut{m/s}}{1.34\ut{m/s^2} } \\&=20 \sqrt{\frac{403}{67}}\ut{s} \\&\approx 49.05068958999181\ut{s} \\&\approx 49.1\ut{s} \end{aligned}$$
(c) $t_{2 \to 3} = 20\ut{s},$ $$\begin{aligned} \\ \bar v_{0 \to 3} &=? \\&= \frac{x_{0 \to 3}}{t_{0 \to 3}} \\&= \frac{x_{0 \to 2}}{t_{0 \to 2}+20\ut{s}} \\&= \frac{806\ut{m} }{20 \sqrt{\frac{403}{67}}\ut{s}+20\ut{s}} \\&= \frac{403 \left(\sqrt{27001}-67\right)}{3360}\ut{m/s} \\&\approx 11.67258436933585\ut{m/s} \\&\approx 11.7\ut{m/s} \end{aligned}$$
(d) $$(t \le t_1)$$ $$\begin{aligned} \Delta x &= v_0t+\frac{1}{2}at^2, \\ x(t)-x_0&= v_0(t-t_0)+\frac{1}{2}a_{0\to 1}(t-t_0)^2 \\ x(t)&= \frac{1}{2}a_{0\to 1}t^2 \\ &= \frac{1}{2}(1.34)t^2 \\ &= \frac{67}{100}t^2 \end{aligned}$$ $$\begin{aligned} v(t)&=\dot x = \dxt{x} \\&=\dt\left(\frac{67}{100}t^2\right) \\&=\frac{67}{50}t \end{aligned}$$ $$\begin{aligned} a(t)&=\dot v = \dxt{v} \\&=\dt\left(\frac{67}{50}t\right) \\&=\frac{67}{50} \end{aligned}$$ $$(t_1 \le t \le t_2)$$ $$\begin{aligned} \Delta x &= vt-\frac{1}{2}at^2, \\ x_2-x(t) &= v_2(t_2-t)-\frac{1}{2}(a_{1\to 2})(t_2-t)^2 \\ x(t) &= x_2-\frac{1}{2}a_{0\to 1}(t_2-t)^2 \\ &= 806-\frac{1}{2}(1.34)\left(20 \sqrt{\frac{403}{67}}-t\right)^2 \\ &= -\frac{67}{100}t^2+\frac{2 \sqrt{27001}}{5}t-806 \end{aligned}$$ $$\begin{aligned} v(t)&=\dot x = \dxt{x} \\&=\dt\left(-\frac{67}{100}t^2+\frac{2 \sqrt{27001}}{5}t-806\right) \\&=\frac{2 \sqrt{27001}}{5}-\frac{67}{50}t \end{aligned}$$ $$\begin{aligned} a(t)&=\dot v = \dxt{v} \\&=\dt\left(-\frac{67}{50}t\right) \\&=-\frac{67}{50} \end{aligned}$$ $$(t_2 \le t \le t_2+20\ut{s})$$ $$\begin{aligned} x(t) &= x_2 = 806 \\ v(t)&=\dot x = \dxt{x} = 0 \\ a(t)&=\dot v = \dxt{v} = 0 \end{aligned}$$ $$(\text{Answers})$$ $$\begin{cases} t_1&=10 \sqrt{\frac{403}{67}}\ut{s} \\ t_2&=20 \sqrt{\frac{403}{67}}\ut{s} \end{cases}$$ $$x(t)=\begin{cases} \frac{67}{100}t^2, &(t \le t_1) \\ -\frac{67}{100}t^2+\frac{2 \sqrt{27001}}{5}t-806, &(t_1\le t\le t_2) \\ 806, &(t_2\le t\le t_2+20\ut{s}) \end{cases}$$ $$v(t)=\begin{cases} \frac{67}{50}t, &(t \le t_1) \\ \frac{2 \sqrt{27001}}{5}-\frac{67}{50}t, &(t_1 \le t \le t_2) \\ 0, &(t_2 \le t \le t_2+20\ut{s}) \end{cases}$$ $$a(t)=\begin{cases} \frac{67}{50}, &(t \le t_1) \\ -\frac{67}{50}, &(t_1 \le t \le t_2) \\ 0, &(t_2 \le t \le t_2+20\ut{s}) \end{cases}$$