Motion in Two and Three Dimensions & Projectile Motion

Vocabulary flashcards covering displacement, velocity, acceleration in two and three dimensions, and kinematic formulas for projectile motion.

Position vector ($\vec{r}$)

A vector used to specify the position of a particle in space, expressed in unit vector notation as $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$.

Vector components of $\vec{r}$

The directional parts of the position vector along the coordinate axes, denoted as $x\hat{i}$, $y\hat{j}$, and $z\hat{k}$.

Scalar components of $\vec{r}$

The coordinate coefficients $x$, $y$, and $z$ that describe the magnitudes of the vector components of $\vec{r}$.

Displacement ($\Delta\vec{r}$)

The change in a particle's position vector from $\vec{r}_1$ to $\vec{r}_2$ during a time interval, given mathematically by $\Delta\vec{r} = \vec{r}_2 - \vec{r}_1$.

Average velocity ($\vec{v}_{avg}$)

The ratio of a particle's displacement $\Delta\vec{r}$ to the time interval $\Delta t$, expressed as $\vec{v}_{avg} = \frac{\Delta\vec{r}}{\Delta t} = \frac{\Delta x}{\Delta t}\hat{i} + \frac{\Delta y}{\Delta t}\hat{j} + \frac{\Delta z}{\Delta t}\hat{k}$.

Instantaneous velocity ($\vec{v}$)

The first time derivative of the position vector, representing the velocity of a particle at a specific instant: $\vec{v} = \frac{d\vec{r}}{dt} = \frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j} + \frac{dz}{dt}\hat{k} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$.

Average acceleration ($\vec{a}_{avg}$)

The change in velocity from $\vec{v}_1$ to $\vec{v}_2$ over a time interval $\Delta t$, expressed as $\vec{a}_{avg} = \frac{\vec{v}_2 - \vec{v}_1}{\Delta t} = \frac{\Delta\vec{v}}{\Delta t}$.

Instantaneous acceleration ($\vec{a}$)

The limit of the average acceleration as the time interval approaches zero, given by $\vec{a} = \lim_{\Delta t \to 0} \frac{\Delta\vec{v}}{\Delta t} = \frac{d\vec{v}}{dt} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$.

Equations of motion for constant acceleration

The kinematic formulas ($v = v_o + at$, $s = v_ot + \frac{1}{2}at^2$, $v^2 = v_o^2 + 2as$, $s = \frac{1}{2}(v_o + v)t$, and $s = vt - \frac{1}{2}at^2$) that apply exclusively when acceleration is uniform.

Projectile

A particle that moves in a vertical plane with an initial velocity $\vec{v}_o$ and experiences an acceleration equal solely to the downward free-fall acceleration $\vec{g}$.

Projectile motion

Motion in a vertical plane where horizontal motion (with constant velocity and zero acceleration) and vertical motion (with constant downward acceleration $-g$) proceed independently of one another.

Maximum height ($H$)

The peak vertical displacement of a projectile above its launch point, occurring when the vertical velocity component is zero ($v_y = 0$), defined as $H = \frac{v_o^2\sin^2(\theta)}{2g}$.

Time to maximum height ($t$)

The duration of time required for a projectile launched at an angle $\theta$ to reach its peak altitude, given by $t = \frac{v_o\sin(\theta)}{g}$.

Total time of flight ($T$)

The total time a projectile remains in the air before landing at the same horizontal level ($H = 0$), given by $T = \frac{2v_o\sin(\theta)}{g} = 2t$.

Horizontal range ($R$)

The total horizontal displacement covered by a projectile over its time of flight, given by $R = \frac{v_o^2\sin(2\theta)}{g}$.

Maximum horizontal range ($R_{max}$)

The greatest horizontal range achievable for a given launch speed $v_o$, given by $R_{max} = \frac{v_o^2}{g}$, which occurs at a launch angle of $\theta = 45^\circ$.