Unit 1 Kinematics: Motion in Two Dimensions (AP Physics C: Mechanics)

Vector

A quantity with both magnitude and direction (e.g., displacement, velocity, acceleration).

Scalar

A quantity with magnitude only (e.g., time, mass, speed, distance).

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

The change in position; equals final position minus initial position ($\textbf{r}_f - \textbf{r}_i$).

Velocity (\textbf{v})

The rate of change of position with time; $\textbf{v} = \frac{d\textbf{r}}{dt}$.

Acceleration ($\textbf{a}$)

The rate of change of velocity with time; $\textbf{a} = \frac{d\textbf{v}}{dt} = \frac{d^2\textbf{r}}{dt^2}$.

Component form of a vector

Writing a 2D vector as $\textbf{A} = A_x \textbf{i} + A_y \textbf{j}$, separating it into x- and y-components.

Unit vectors ($\textbf{i}$ and $\textbf{j}$)

Magnitude-1 vectors that point along the +x direction ($\textbf{i}$) and +y direction ($\textbf{j}$).

Vector components ($A_x, A_y$)

Signed projections of a vector onto the coordinate axes; can be interpreted as “shadow lengths” on x and y axes.

Magnitude from components

The size of a vector found by |\textbf{A}| = \textbf{√(A_x^2 + A_y^2)}.

Direction angle from components

An angle θ (measured CCW from +x) related by tanθ = Ay/Ax; must check the correct quadrant using signs of Ax and Ay.

Components from magnitude and angle

For magnitude A at angle $\theta$ above +x: $A_x = A \cos(\theta)$ and $A_y = A \sin(\theta)$ (with signs consistent with the chosen angle/quadrant).

Vector addition (component-wise)

Adding vectors by adding corresponding components: $C_x = A_x + B_x$ and $C_y = A_y + B_y$.

Vector subtraction

Subtracting by adding the negative: ⃗A − ⃗B = ⃗A + (−⃗B).

Position vector in 2D ($\textbf{r}(t)$)

A time-dependent position written as $\textbf{r}(t) = x(t)\textbf{i} + y(t)\textbf{j}$.

Independence of x and y motion (2D kinematics)

After breaking vectors into components, you can apply 1D kinematics separately along x and y and recombine results.

Distance vs. displacement

Distance is total path length (a scalar); displacement is net change in position (a vector) and can have a smaller magnitude than distance traveled.

Projectile motion (model)

2D motion where the only acceleration is gravity (near Earth), air resistance is neglected, and motion splits into independent horizontal and vertical parts.

Projectile acceleration components

For ideal projectile motion: $a_x = 0$ and $a_y = -g$ (with +y upward by convention).

Projectile horizontal equations

With $a_x = 0: v_x(t) = v_{0x} \text{ (constant)}$ and $x(t) = x_0 + v_{0x} t$.

Projectile vertical equations

With $a_y = -g: v_y(t) = v_{0y} - gt$ and $y(t) = y_0 + v_{0y} t - \frac{1}{2}gt^2$.

Top of a projectile’s path

The point where the vertical velocity component is zero ($v_y = 0$), while the horizontal component $v_x$ generally remains nonzero.

Symmetric (level-ground) projectile results

If launch and landing heights are the same: $t_{\text{top}} = \frac{v_{0y}}{g}$, \Deltay_{\max} = \frac{v_{0y}^2}{2g}, and $t_{\text{flight}} = \frac{2v_{0y}}{g}$.

Projectile range formula (level ground)

For same launch/landing height with no air resistance: $R = \frac{(v_0^2 \sin(2\theta))}{g}$.

Trajectory equation (parabolic path)

Eliminating time using $t = \frac{x}{v_{0x}}$ gives $y = \frac{v_{0y}}{v_{0x}}x - \frac{1}{2}g\bigg(\frac{x}{v_{0x}}\bigg)^2$, which is quadratic in x (a parabola).

Relative velocity notation ($\textbf{v}_{A/B}$)

$\textbf{v}_{A/B}$ means “velocity of A as measured from B,” with the relationship $\textbf{v}_{A/C} = \textbf{v}_{A/B} + \textbf{v}_{B/C}$ (Galilean relativity).