General Physics I:

\begin{align*} \vec{r}(2s)\amp = \left(A\cdot 2^{3}+B \cdot 2^{2}\right)\hat{i}+C\cdot 2\hat{j}\\ \amp = \left(3\cdot 2^{3}+2 \cdot 2^{2}\right)\hat{i}-1\cdot 2\hat{j} \\ \amp = 32\hat{i}-2\hat{j}. \end{align*}

\begin{align*} v_{avg}(0,2s) \amp = \frac{\vec{r}(2s)-\vec{r}(0)}{2-0}\\ \amp = \frac{32\hat{i}-2\hat{j}-0}{2}\\ \amp =16\hat{i}-\hat{j} \end{align*}

\begin{align*} \vec{v}(-1s) \amp = \frac{\,d\vec{r}}{\,dt} \\ \amp = \left(3At^{2}+2Bt\right)\hat{i}+C\hat{j} \\ \amp = \left[3.3.(-1)^{2}+2.2.(-1)\right]\hat{i}+(-1)\hat{j} \\ \amp = 5\hat{i}-\hat{j} \end{align*}

\begin{align*} \vec{v}(3s) \amp = \left(3At^{2}+2Bt\right)\hat{i}+C\hat{j} \\ \amp = \left[3.3.(3)^{2}+2.2.(3)\right]\hat{i}+(3)\hat{j} \\ \amp = 93\hat{i}-\hat{j} \end{align*}

\begin{align*} a_{avg}(-1s,3s) \amp = \frac{v(3s)-v(-1s)}{3-(-1)} \\ \amp = \frac{93\hat{i}-\hat{j}-5\hat{i}+\hat{j}}{4} \\ \amp = 22\hat{i} \end{align*}

\begin{align*} \vec{a}(0.5s)\amp = \frac{\,dv}{\,dt} \\ \amp = \left(6At+2B\right)\hat{i} \\ \amp = [6.3.(0.5) +2.2]\hat{i} \\ \amp =13\hat{i} \end{align*}

\begin{align*} \vec{r}(0) \amp = r \cos\omega 0\hat{i}+ r \sin \omega 0\hat{j} \\ \amp = r \hat{i}= 1 m \hat{i} \end{align*}

\begin{align*} \vec{r}(1s) \amp = r \cos\omega t\hat{i}+ r \sin \omega t\hat{j} \\ \amp = r \cos\pi \hat{i}+ r \sin \pi\hat{j} = -1m\hat{i} \end{align*}

\begin{align*} \vec{v}_{av}(0,1s)\amp =\frac{r_{1}-r_{o}}{t_{2}-t_{1}} \\ \amp = \frac{-1-1}{1-0} =-2 m/s \end{align*}

\begin{align*} \vec{v}(2 s) \amp = \frac{\,d\vec{r}}{\,dt} \\ \amp = -\omega r \sin\omega t\hat{i}+ \omega r \cos \omega t\hat{j} \\ \amp = \pi \cos 2\pi j= \pi \hat{j}= 3.14 m/s \hat{j} \end{align*}

\begin{align*} \vec{a}_{av}(0,2s) \amp =\frac{v_{2}-v_{o}}{t_{2}-t_{1}} = 2\pi m/s \end{align*}

\begin{align*} a(0.5 s) \amp = \frac{\,dv}{\,dt} \\ \amp = -\omega^{2} r \cos\omega t\hat{i} - \omega^{2} r \sin \omega t\hat{j}\\ \amp = -\pi^{2} m/s^{2} \cos\pi/2\hat{i} - \pi^{2} m/s^{2} \sin \pi/2\hat{j}\\ \amp = -\pi^{2} m/s^{2}\hat{j} = - 9.87 m/s^{2}\hat{j} \end{align*}

\begin{align*} a_{r}(1.5 s) \amp = \frac{v^{2}}{r} = \pi^{2} m/s^{2} = 9.87 m/s^{2} \end{align*}

\begin{align*} a_{t}(1.5 s) \amp = r\alpha=r\frac{\,d\omega}{\,dt} = 0 \end{align*}

\begin{align*} a(1.5 s) \amp = \sqrt{a^{2}_{r}+a^{2}_{t}} = \pi m/s^{2} = 3.14 m/s^{2} \end{align*}

\begin{align*} v \amp =r\omega = r\frac{2\pi}{T} \\ \amp = \frac{2\times 5.10\times10^{6}\times \pi}{86164} = 372m/s \end{align*}

\begin{align*} a \amp = \frac{v^{2}}{r} = \frac{372^{2}}{5.10\times 10^{6}} = 2.7\times10^{-2}m/s^{2} \end{align*}

\begin{align*} v_{ox} \amp = v_{o} \cos\theta_{o} = 30\times\cos30^{o}=26 m/s \end{align*}

\begin{align*} v_{oy}\amp = v_{o} \sin\theta_{o} =30\times\sin30^{o}=15 m/s \end{align*}

\begin{align*} x \amp = v_{ox} t \Rightarrow \quad t=\frac{x}{v_{ox}} = \frac{50 m}{26 m/s} = 1.92 s \end{align*}

\begin{align*} y \amp = v_{oy} t -\frac{1}{2} g t^{2} = 15 \times 1.92 -\frac{1}{2}\times9.8\times(1.92)^{2} =10.7 m \end{align*}

\begin{align*} v_{x} \amp = v_{ox} = 26 m/s; \\ v_{y} \amp = v_{oy} - g t = -3.82 m/s;\\ v \amp =\sqrt{v^{2}_{x} + v^{2}_{y}}=26.27 m/s \end{align*}

\begin{align*} v_{ox} \amp = vo \cos\theta_{o};\qquad \theta_{o} = 60^{o}; \qquad v_{oy} = vo \sin\theta_{o};\\ x \amp = 40 yd = 120 ft; \quad y_i=0,\quad y_f=0\\ \text{But,} \quad x \amp = v_{ox} t; \qquad \text{and}\quad g = 32 ft/s^{2} \\ y_f-y_i \amp = v_{oy} t - \frac{1}{2} g t^{2};\\ 0 \amp= vo \sin\theta_{o}\left( \frac{x}{vo \cos\theta_{o}}\right)- \frac{1}{2}\times 32 \left(\frac{x}{vo \cos\theta_{o}}\right)^{2}\\ \therefore \quad v_{o} \amp = 62.3 ft/s \end{align*}

\begin{align*} v_{y}\amp = v_{oy} - g t;\\ v_{y} \amp = 0 \\ t \amp = 1.68 s; \\ y \amp = v_{oy} t - \frac{1}{2}g t^{2} = 45.48 ft \end{align*}

\begin{align*} x \amp = v t; \quad t = 3.85 s\\ x = 60 yd - 40 yd = 20 yd = 60 ft \amp = v t \\ \therefore v \amp = \frac{60}{3.85} =15.58 ft/s \end{align*}

\begin{align*} v^{2}_{f} \amp =v^{2}_{i}-2gh \\ \text{or,} \quad 0 \amp = v^{2}_{oy}-2\times 9.8 \times h\\ \therefore h \amp =11.48 m \end{align*}

\begin{align*} v^{2}_{y} \amp = v^{2}_{oy}-2gy \\ \amp =(15)^{2}-2\times 9.8 \times (-150) = 225+2940 \\ \text{or,}\quad v_{y} \amp =\sqrt{3165}= 56.26 m/s\quad (downward) \\ \therefore v_{f}\amp =\sqrt{v^{2}_{x}+v^{2}_{y}}=\sqrt{(25.98)^{2}+(56.26)^{2}} =61.97 m/s\\ \phi \amp = \tan^{-1}\left( \frac{v_{y}}{v_{x}}\right)= 65.21^{o} \quad\text{[below ground]} \end{align*}

\begin{align*} x \amp =v_{ox}t = 25.98\times 7.27 = 188.87 m \\ \because y_{f}-y_{o}\amp =v_{oy}t-\frac{1}{2}gt^{2}\\ 0-150 \amp =15 t-4.9\times t^{2} \\ \therefore \quad t \amp = -4.21s, 7.27s \end{align*}

\begin{equation*} \theta = \tan^{-1}\left(\frac{y}{x}\right)=37^{o} \end{equation*}

\begin{equation*} x =v_{ox}t =v_{o}\cos\theta t \end{equation*}

\begin{equation*} \therefore \quad t= \frac{x}{v_{o}\cos\theta}=1.67s \end{equation*}

\begin{equation*} h=v_{oy}t+\frac{1}{2}gt^{2}=v_{o}\sin\theta \times (1.67)-\frac{1}{2}\times 9.8\times (1.67)^{2} \end{equation*}

\begin{equation*} \therefore h= 15\times\sin37^{o}\times 1.67-4.9\times (1.67)^{2} =1.4 m \end{equation*}

\begin{align*} h\amp =v_{yo}t+\frac{1}{2}gt^{2}\\ 0 \amp =3.66t-\frac{1}{2}(9.8)t^{2}\\ \therefore \quad t \amp = \frac{2\times3.66}{9.8}=0.747 s\\ and \quad x \amp =v_{xo}t = 0.662\times0.747 = 0.494m \end{align*}

\begin{align*} y_{f} \amp =y_{o}+v_{i}t+\frac{1}{2}gt^{2} \end{align*}

\begin{equation*} \text{or,}\quad y = 0 + v_{yo}\left[ \frac{x}{v_{xo}}\right]-\frac{1}{2}(9.8) \left[\frac{x}{v_{xo}}\right]^{2} \quad [\because \quad x = v_{xo}t] \end{equation*}

\begin{equation*} \therefore y = 5.53 x - (11.2/m) x^{2} \end{equation*}