솔루션피아

$$\begin{cases} h &= 1.50\ut{m}\\ \Delta x &= 1.52\ut{m}\\ \vec a &= -g\j\\ g &\approx 9.80665\ut{m/s^2}\\ \end{cases}$$ (a) $t=?$ $$\begin{aligned} \Delta y &= v_{y0}t+\frac{1}{2}a_yt^2,\\ -1.5 &= (0)t+\frac{1}{2}(-g)t^2\\ 1.5 &= \frac{1}{2}gt^2\\ \end{aligned}$$ $$\begin{aligned} t&=\sqrt{\frac{3}{g}}\\ &\approx 0.553095709524\ut{sec}\\ &\approx 0.553\ut{sec}\\ \end{aligned}$$ (b) $v_0=?$ $$\b..

$$\begin{cases} \vec v_0 &= 6.75\i\ut{m/s}\\ \Delta \vec r &= -5.00\j\ut{cm}\\ \vec a &= -g\j\\ g &\approx 9.80665\ut{m/s^2}\\ \end{cases}$$ $$ \Delta x =? $$ $$\begin{aligned} \Delta y &= v_{y0}t+\frac{1}{2}a_yt^2,\\ -5.00\times10^{-2} &= 0\cdot t+\frac{1}{2}(-g)t^2\\ 5.00\times10^{-2} &= \frac{1}{2}gt^2\\ t&= \frac{1}{\sqrt{10g}} \end{aligned}$$ $$\begin{aligned} \Delta x &= v_x t = v_0 t\\ &=..

$$\begin{cases} A_y&=30\ut{m}\\ \vec v_A&=3.0\i\ut{m/s}\\ \vec v_{B0}&=0\\ \abs{\vec a_{B}}&=0.40\ut{m/s^2}\\ \end{cases} $$ $$ \begin{aligned} \vec r_A &=\Delta \vec r_A + \vec r_{A0}\\ &=\int \vec v_A\dd t+\vec r_{A0}\\ &=\int 3\i\dd t+30\j\ut{m}\\ &=3t\i+30j\ut{m}\\ \end{aligned} $$ $$ \vec a_B = \frac{2}{5}\sin\theta\i+\frac{2}{5}\cos\theta\j\ut{m/s^2}$$ $$ \begin{aligned} \vec v_B&=\vec v_{..

$$\begin{cases} \vec a&=3t\i+4t\j\ut{m/s^2}\\ \vec r_0&=20.0\i+40.0\j\ut{m}\\ \vec v_0&=5.00\i+2.00\j\ut{m/s}\\ \end{cases} $$ $$ \begin{aligned} \vec v&=\Delta \vec v+\vec v_0\\ &=\int_0^t \vec a\dd t+\vec v_0\\ &=\int_0^t 3t\i+4t\j\dd t+\vec v_0\\ &=\(\frac{3}{2}t^2+5\)\i+\(2 t^2+2\)\j\\ \end{aligned} $$ $$ \begin{aligned} \vec r&=\Delta \vec r+\vec r_0\\ &=\int_0^t \vec v\dd t+\vec r_0\\ &=\i..

$$\begin{cases} \vec a&=5.00\i+7.00\j\ut{m/s^2}\\ \vec v_0&=4.00\i\ut{m/s}\\ \end{cases} $$ $$t_{\Delta x=10}=?$$ $$ \begin{aligned} \Delta r_x &= v_{x0}t+\frac{1}{2}a_xt^2\\ 10 &= 4t+\frac{1}{2}\cdot5t^2\\ t_{\Delta x=10}&=\frac{2}{5} \left(\sqrt{29}-2\right) \end{aligned} $$ $$v_{\Delta x=10}=?$$ $$ \begin{aligned} \vec v&=\vec v_0 + \vec a t,\\ \vec v_{\Delta x=10}&=\vec v_0 + \vec a t_{\Delt..

$$\begin{cases} \vec a&=6.0\i-3.0\j\ut{m/s^2}\\ \vec v_0&=12.0\i+18.0\j\ut{m/s}\\ \end{cases} $$ $$ \begin{aligned} \vec v&=\vec v_0+\vec a t\\ &=12.0\i+18.0\j+(6.0\i-3.0\j)t\ut{m/s}\\ &=(12.0+6.0t)\i+(18.0-3.0t)\j\ut{m/s}\\ \end{aligned} $$ $$v_y=18.0-3.0t=0$$ $$t=6\ut{s}$$ $$ \begin{aligned} \vec v_6&=(12.0+6.0\cdot6)\i+(18.0-3.0\cdot6)\j\ut{m/s}\\ &=48\i\ut{m/s} \end{aligned} $$

$$\vec v = (6.0t-4.0t^2)\i+8.0\j\ut{m/s}$$ (a) $\vec a_{2.5}=?$ $$\begin{aligned} \vec a &= \dxt{\vec v}\\ &= \dt\{(6.0t-4.0t^2)\i+8.0\j\}\\ &= (6-8t)\i\ut{m/s^2}\\ \end{aligned}$$ $$ \begin{aligned} \vec a_{t=2.5}&=(6-8\cdot2.5)\i\ut{m/s^2}\\ &=-14\i\ut{m/s^2} \end{aligned} $$ (b) $t_{a=0}=?$ $$\vec a = (6-8t)\i=0$$ $$ \begin{aligned} t&=\frac{3}{4}\ut{s}\\ &= 0.75\ut{s} \end{aligned} $$ (c) $t..

$$\begin{cases} \vec v_0 &= 7.0\i\ut{m/s}\\ \vec a &= -9.0\i+3.0\j\ut{m/s^2}\\ \end{cases}$$ $$\begin{aligned} \vec v &= \vec v_0+\vec at\\ &= 7.0\i+(-9.0\i+3.0\j)t\ut{m/s}\\ &= (7.0-9.0t)\i+3.0t\j\ut{m/s}\\ \end{aligned}$$ $$\vec v_x = 7.0-9.0t=0,$$ $$t_{r_{xM}} = \frac{7}{9}\ut{s}$$ (a) $v_{r_{xM}}=?$ $$\begin{aligned} v_{r_{xM}} &= v_{\frac{7}{9}}\\ &=\(7.0-9.0\cdot\frac{7}{9}\)\i+3.0\cdot\fr..

$$\begin{cases} \vec v_0 &= 4.0\i-2.0\j+3.0\k\ut{m/s}\\ \vec v_{4.0} &= -2.0\i-2.0\j+5.0\k\ut{m/s}\\ \end{cases}$$ (a) $\vec a_{avg}=?$ $$\begin{aligned} \vec a_{avg} &= \frac{\Delta \vec v}{\Delta t}\\ &=\frac{\vec v_4-\vec v_0}{4}\\ &=\frac{(-2.0\i-2.0\j+5.0\k)-(4.0\i-2.0\j+3.0\k)}{4}\ut{m/s^2}\\ &=-1.5\i+0.5\k\ut{m/s^2}\\ \end{aligned}$$ (b) $\abs{\vec a_{avg}}=?$ $$\begin{aligned} \abs{\vec ..

$$\vec r=\i+3t^4\j+ t\k\ut{m}$$ (a) $$\begin{aligned} \vec v &= \dxt{\vec r}\\ &=\dt(\i+3t^4\j+ t\k)\\ &=12t^3\j+\k\ut{m/s}\\ \end{aligned}$$ (b) $$\begin{aligned} \vec a &= \dxt{\vec v}\\ &=\dt(12t^3\j+\k)\\ &=36t^2\j\ut{m/s^2}\\ \end{aligned}$$