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# Chapter 3. Motion in Two or Three Dimensions ## 3.1 Position and Velocity ### Position Vector $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ ### Displacement $\Delta\vec{r} = \vec{r_2} - \vec{r_1}$ ### Average Velocity $\vec{v}_{avg} = \frac{\Delta\vec{r}}{\Delta t}$ ### Instantaneous Velocity...

# Chapter 3. Motion in Two or Three Dimensions ## 3.1 Position and Velocity ### Position Vector $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ ### Displacement $\Delta\vec{r} = \vec{r_2} - \vec{r_1}$ ### Average Velocity $\vec{v}_{avg} = \frac{\Delta\vec{r}}{\Delta t}$ ### Instantaneous Velocity $\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta\vec{r}}{\Delta t} = \frac{d\vec{r}}{dt}$ $\vec{v} = \frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j} + \frac{dz}{dt}\hat{k}$ ### Average Speed $v_{avg} = \frac{\text{total distance}}{\Delta t}$ ## 3.2 Acceleration ### Average Acceleration $\vec{a}_{avg} = \frac{\vec{v_2} - \vec{v_1}}{\Delta t} = \frac{\Delta\vec{v}}{\Delta t}$ ### Instantaneous Acceleration $\vec{a} = \lim_{\Delta t \to 0} \frac{\Delta\vec{v}}{\Delta t} = \frac{d\vec{v}}{dt}$ $\vec{a} = \frac{dv_x}{dt}\hat{i} + \frac{dv_y}{dt}\hat{j} + \frac{dv_z}{dt}\hat{k}$ $\vec{a} = \frac{d^2x}{dt^2}\hat{i} + \frac{d^2y}{dt^2}\hat{j} + \frac{d^2z}{dt^2}\hat{k}$ ## 3.3 Projectile Motion ### Assumptions * $a_x = 0$ * $a_y = -g = -9.8 m/s^2$ * ignore air resistance ### Kinematic Equations | | x-component | y-component | | :----------------------------------------------------- | :------------------------------------------ | :------------------------------------------------- | | **Velocity** | $v_x = v_{0x} = v_0cos\theta_0$ | $v_y = v_{0y} - gt = v_0sin\theta_0 - gt$ | | **Position** | $x = x_0 + v_{0x}t = x_0 + v_0cos\theta_0t$ | $y = y_0 + v_{0y}t - \frac{1}{2}gt^2 = y_0 + v_0sin\theta_0t - \frac{1}{2}gt^2$ | | **Velocity as a function of position** | $v_x = v_{0x} = v_0cos\theta_0$ | $v_y^2 = v_{0y}^2 - 2g(y - y_0) = (v_0sin\theta_0)^2 - 2g(y - y_0)$ | | **Horizontal Range (With $\Delta y = 0 = y - y_0$)** | $R = \frac{v_0^2}{g}sin2\theta_0$ | | ### Trajectory $y = (tan\theta_0)x - (\frac{g}{2(v_0cos\theta_0)^2})x^2$ (Equation of a parabola) ### Projectile motion is made up of * uniform motion along the horizontal direction * constant acceleration motion along the vertical direction ## 3.4 Uniform Circular Motion ### Period $T = \frac{2\pi r}{v}$ ### Radial Acceleration $a_{rad} = \frac{v^2}{r}$ ### Frequency $f = \frac{1}{T}$ ### Speed $v = \frac{2\pi r}{T} = 2\pi r f$ ### Uniform circular motion * The speed is constant * Uniform means constant ## 3.5 Tangential and Radial Acceleration $\vec{a} = \vec{a}_{rad} + \vec{a}_{tan}$ ### Radial (centripetal) acceleration $a_{rad} = \frac{v^2}{r}$ ### Tangential acceleration $a_{tan} = \frac{dv}{dt}$ ### Total Acceleration $a = \sqrt{a_{tan}^2 + a_{rad}^2}$ ## 3.6 Relative Motion ### Reference Frames $\vec{r}_{PA} = \vec{r}_{PB} + \vec{r}_{BA}$ ### Relative Velocities $\vec{v}_{PA} = \vec{v}_{PB} + \vec{v}_{BA}$ Where: * P = particle * A = reference frame A * B = reference frame B **Note:** The order of the subscripts matters!!!