Page Description Equation
719 vector between two points. The vector $\vect{v}=\overrightarrow{PQ}$ between two points $P_1\left(x_1,y_1\right)$ and $P_2\left(x_2,y_2\right)$ \[ v=\left(x_2-x_1,y_2-y_1\right) \]
725 vector tangent to a curve at a point $x=c$ \[ \vect{v}=\left(a,b\right)\\ \text{where}\\ y=f\left(x\right)\text{ and } \left.\frac{dy}{dx}\right|_{x=c}=\frac{b}{a} \]
725 vector normal to a vector $\vect{u}$. If $\vect{n}$ is a vector normal to $\vect{u}$ then so is $-\vect{n}$ and the implication holds. \[ \vect{u}=\left(a,b\right)\Rightarrow\vect{n} =\left(-b,a\right)\land-\vect{n} =\left(b,-a\right) \]
725 vector normal to a curve at a point $P_0.$ If $\vect{v}=\left(a,b\right)$ is a vector tangent to a curve at $x=c$ then $\vect{n}$ and $-\vect{n}$ are vectors normal to the curve at that point. \[ \vect{n}=\left(-b,a\right) \text{ and } -\vect{n}=\left(b,-a\right) \]
721 vector components from angle with $x$-axis. $\vect{v}$ forms an angle $\theta$ with the $x$-axis (or polar axis) \[ \vect{v}= \left( \abs{\vect{v}} \cos\theta, \abs{\vect{v}} \sin{\theta} \right) \]
737 equation of line perpendicular to a vector. The line and the vector are perpendicular. \[ Ax+By=C\\ \vect{n}=A\vect{i}+B\vect{j} \]
737 equation of line parallel to a vector. The line and the vector are parallel. \[ Bx-Ay=C\\ \vect{n}=A\vect{i}+B\vect{j} \]
729 angle between vectors \[ \theta=\cos^{-1} \frac{\vect{u}\cdot\vect{v}}{\abs{\vect{u}}\abs{\vect{v}}} \]
733 projection of $\vect{u}$ onto $\vect{v}$ \[ \text{proj}_\vect{v}\vect{u} = \left( \frac{\vect{u}\cdot\vect{v}} {\abs{\vect{v}}^2} \right) \vect{v} \]
733 scalar component of $\vect{u}$ in direction of $\vect{v}.$ If $\theta$ is the angle between $\vect{u}$ and $\vect{v}$ then the scalar component of $\vect{u}$ in the direction of $\vect{v}$ is defined as given. \[ \abs{\vect{u}}\cos\theta =\frac{\vect{u}\cdot\vect{v}}{\abs{\vect{v}}} \]
738 position vector \[ \vect{r}\left(t\right) =f\left(t\right)\vect{i}+g\left(t\right)\vect{j} \]
743 velocity vector \[ \vect{v}\left(t\right) =\vect{r}' =\frac{d\vect{r}}{dt} =\abs{\vect{v}}\frac{\vect{v}}{\abs{\vect{v}}} \]
743 speed \[ \abs{\vect{v}\left(t\right)} \]
743 acceleration vector \[ \vect{a}\left(t\right) =\vect{v}'\left(t\right) =\frac{d\vect{v}}{dt} =\vect{r}'' =\frac{d^2\vect{r}}{dt^2} =\abs{\vect{a}}\frac{\vect{a}}{\abs{\vect{a}}} \]
743 smooth. If $\vect{r}'\left(t\right)=\zeros$ then $\vect{r}$ is not smooth. \[ \]
743 unit vector in direction of motion \[ \vect{T}=\frac{\vect{v}}{\abs{\vect{v}}} \]
739 spiral of archimedes \[ \vect{r}\left(t\right) =\left(t\cos t\right)\vect{i} +\left(t\sin t\right)\vect{j}, \quad t\gt0 \]
initial vertical velocity \[ v_{y_i}=v_0\sin\alpha \]
initial horizontal velocity \[ v_{x_i}=v_0\cos\alpha \]
initial velocity \[ \begin{align*} \vect{v}_0 &=v_{x_i}\vect{i}+v_{y_i}\vect{j}\\ &=\left(v_0\sin\alpha\right)\vect{i} +\left(v_0\cos\alpha\right)\vect{j} \end{align*} \]
position vector (vector equation for ideal projectile motion) and displacement vector \[ \Delta\vect{r}=\vect{r}-\vect{r}_0=\vect{v}_0+\frac{1}{2}\vect{g}t^2\\ \vect{r}=\vect{r}_0+\vect{v}_0+\frac{1}{2}\vect{g}t^2 \]
horizontal displacement \[ \begin{align*} x &=x_0+v_{x_i}t\\ \Delta x &=x-x_0\\ &=v_{x_i}t\\ &=\left(v_0\cos\alpha\right)t \end{align*} \]
vertical displacement \[ \begin{align*} y &=y_0+v_{y_i}t-\frac{1}{2}gt^2\\ \Delta y &=y-y_0\\ &=v_{y_i}t-\frac{1}{2}gt^2\\ &=\left(v_0\sin\alpha\right)t -\frac{1}{2}gt^2 \end{align*} \]
launch angle (firing angle, angle of elevation) \[ \alpha=\tan^{-1}\frac{v_y}{v_x} \]
time when maximum height is reached \[ t_\text{halfway}=\frac{v_0\sin\alpha}{g} \]
flight time \[ t_f=2\cdot\frac{v_0\sin\alpha}{g} \]
maximum height \[ y_\text{max}=\frac{\left(v_0\sin\alpha\right)^2}{2g} \]
range \[ R=\frac{v_0^2}{g}\sin 2\alpha \]
773 area between origin and a polar curve \[ \]
775 area between two polar curves \[ \]

Notes

Page Notes
738 Vector Functions
739 Limits of Vector Functions
739 Continuity of Vector Functions
739 Continuity Test
740 Derivative of a Vector Function (Definition).
741 Differentiation Rules for Vector Functions. Differentiate the components using the differentiation rules for real functions. Use the product rule on dot products. The cross product isn’t defined on 2-space vectors so no derivative is either.
745 Indefinite Integral of a Vector Function. Integrate the components. Constant of integration is a vector.
745 Definite Integral of a Vector Function
741 Direction of $\vect{r}'$: The velocity vector $\vect{r}'$ points in the direction of motion tangent to $\vect{r}.$

Other Notes for Chapter 9.4 Projectile Motion

List of variables and parameters used in projectile motion: \[ t, x, y, x_0, y_0, \Delta x, \Delta y, g, \sin\alpha, \cos\alpha, \vect{v}_0, v_0, v_{x_i}, v_{y_i}, \alpha, v_0\sin\alpha, v_0\cos\alpha, \vect{r}, \vect{r}_0, \Delta \vect{r}, y\text{flight}_\text{max} \] Derivation of the position vector displacement vectors in projectile motion.