# Kinematics in Two Dimensions

> AP Physics C: Mechanics · Unit 1: Kinematics
> Source: https://www.owlsprep.com/study/ap-physics-c-mech-u1-kinematics-in-two-dimensions/

This guide covers core concepts of planar kinematics for AP Physics C: Mechanics, including vector decomposition, 2D constant acceleration kinematics, projectile motion, and relative velocity, aligned to the official College Board CED.

**Prerequisites:** [One-dimensional constant acceleration kinematics](https://www.owlsprep.com/study/ap-physics-c-mech-u1-kinematics-one-dimension/); Basic vector operations and single-variable differentiation/integration

## Learning objectives

- Decompose 2D kinematic vectors into perpendicular components
- Apply constant-acceleration kinematics component-wise to planar motion
- Solve projectile motion problems for unequal launch/landing heights
- Calculate relative velocity in 2D inertial reference frames

## Core Principles of Two-Dimensional Kinematics

Kinematics in two dimensions (also called planar kinematics) is the study of objects moving along any path in a flat plane, tracking position, velocity, and acceleration without referencing the forces that cause motion. This topic contributes 7-10% of your total AP Physics C: Mechanics exam score, appearing in both multiple-choice and free-response sections.

**Two-Dimensional Kinematics** — Study of planar motion where all motion occurs in a 2D coordinate plane. Perpendicular components of motion are independent, allowing separate analysis of x and y motion.

*Notation:* $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}$, $\vec{v}(t) = \frac{d\vec{r}}{dt}$, $\vec{a}(t) = \frac{d\vec{v}}{dt}$

> **info**
>
> The core principle underpinning all 2D kinematics: Perpendicular components of motion are independent of each other, so you can analyze x and y motion separately then combine results.

## Vector Decomposition and General 2D Kinematic Equations

The key to solving 2D kinematics problems is decomposing all vector quantities (position, velocity, acceleration) into perpendicular components, then applying 1D kinematic rules to each component independently. For constant acceleration, integration leads to the vector form of the kinematic equations:

$$\vec{v}(t) = \vec{v}_0 + \vec{a} t \\ \vec{r}(t) = \vec{r}_0 + \vec{v}_0 t + \frac{1}{2}\vec{a} t^2$$

In component form, this gives four separate equations: $v_x(t) = v_{0x} + a_x t$, $v_y(t) = v_{0y} + a_y t$, $x(t) = x_0 + v_{0x}t + \frac{1}{2}a_x t^2$, $y(t) = y_0 + v_{0y}t + \frac{1}{2}a_y t^2$. For non-constant acceleration, differentiate or integrate each component separately, following the same rules as 1D kinematics.

**Worked example:** An object moves in the xy-plane with initial position $\vec{r}_0 = (2\hat{i} + 3\hat{j}) \text{ m}$, initial velocity $\vec{v}_0 = (4\hat{i} - 1\hat{j}) \text{ m/s}$, and constant acceleration $\vec{a} = (-2\hat{i} + 6\hat{j}) \text{ m/s}^2$. Find the speed of the object and its position at $t = 2 \text{ s}$.

1. Separate the x and y components from the given vectors:

   $$x_0 = 2 \text{ m}, v_{0x} = 4 \text{ m/s}, a_x = -2 \text{ m/s}^2 \\ y_0 = 3 \text{ m}, v_{0y} = -1 \text{ m/s}, a_y = 6 \text{ m/s}^2$$
2. Calculate velocity components at $t=2 \text{ s}$:

   $$v_x = 4 + (-2)(2) = 0 \text{ m/s} \\ v_y = -1 + 6(2) = 11 \text{ m/s}$$
3. Speed is the magnitude of the velocity vector:

   $$v = \sqrt{v_x^2 + v_y^2} = \sqrt{0^2 + 11^2} = 11 \text{ m/s}$$
4. Calculate position components for the final position vector:

   $$x = 2 + 4(2) + \frac{1}{2}(-2)(2)^2 = 6 \text{ m} \\ y = 3 + (-1)(2) + \frac{1}{2}(6)(2)^2 = 13 \text{ m} \\ \vec{r} = (6\hat{i} + 13\hat{j}) \text{ m}$$

> **Exam tip:** Always remember that speed (a scalar) is the Pythagorean magnitude of the velocity vector, not the algebraic sum of the x and y components. AP MCQs regularly include trap answers that incorrectly add components instead of taking their magnitude.

## Projectile Motion

Projectile motion is the most common 2D kinematics scenario tested on the AP exam: it describes the motion of an object moving under the influence of constant gravitational acceleration, with air resistance ignored. The standard coordinate system sets $+y$ upward (vertical) and $+x$ horizontal, so acceleration components are $a_x = 0$ and $a_y = -g = -9.8 \text{ m/s}^2$. Because $a_x = 0$, horizontal velocity is constant for the entire flight.

For a projectile launched with initial speed $v_0$ at an angle $\theta$ above the horizontal, the initial velocity components are $v_{0x} = v_0 \cos\theta$ and $v_{0y} = v_0 \sin\theta$, leading to the kinematic equations:

$$x(t) = x_0 + v_0 \cos\theta \cdot t \\ y(t) = y_0 + v_0 \sin\theta \cdot t - \frac{1}{2} g t^2$$

> **tip**
>
> While derived formulas for maximum height, time of flight, and range exist for equal launch/landing heights, the AP exam almost always tests cases where heights differ. Always start from the full equations above rather than memorized shortcuts.

**Worked example:** A projectile is launched from ground level ($y_0=0$) over flat ground, with initial speed $20 \text{ m/s}$ at $30^\circ$ above horizontal. How high is the projectile when it is $20 \text{ m}$ horizontally from the launch point? Use $g=9.8 \text{ m/s}^2$.

1. Calculate initial velocity components:

   $$v_{0x} = 20 \cos30^\circ = 10\sqrt{3} \approx 17.32 \text{ m/s} \\ v_{0y} = 20 \sin30^\circ = 10 \text{ m/s}$$
2. Find the time when $x = 20 \text{ m}$ using constant horizontal velocity:

   $$t = \frac{x}{v_{0x}} = \frac{20}{10\sqrt{3}} = \frac{2}{\sqrt{3}} \approx 1.155 \text{ s}$$
3. Substitute $t$ into the y-position equation to find height:

   $$y = v_{0y} t - \frac{1}{2} g t^2 = 10\left(\frac{2}{\sqrt{3}}\right) - 4.9\left(\frac{4}{3}\right) \approx 5.02 \text{ m}$$

> **Exam tip:** Always confirm the sign of gravitational acceleration at the start of your working. If you set $+y$ upward, $a_y = -g$, not $+g$. A flipped sign here will lead to wrong answers across the entire problem, and FRQ graders will deduct multiple points for this consistent error.

## Relative Motion in Two Dimensions

Relative velocity describes the motion of an object as measured from different inertial reference frames (frames moving at constant velocity relative to each other). The core rule follows vector addition and a simple subscript cancellation rule.

**Relative Velocity Rule** — To find the velocity of A relative to C, add the velocity of A relative to B to the velocity of B relative to C. The inner subscript cancels out. Reversing subscript order flips the sign of the vector.

*Notation:* $\vec{v}_{A/C} = \vec{v}_{A/B} + \vec{v}_{B/C}$, $\vec{v}_{A/B} = -\vec{v}_{B/A}$

**Worked example:** A boater wants to cross a 100 m wide river that flows east at 3 m/s. The boater can row the boat at 5 m/s relative to the water, and aims the boat directly north across the river. What is the boat's speed relative to the ground, and how far downstream does the boater land on the opposite bank?

1. Set up coordinates with $+x$ east and $+y$ north, write known velocities:

   $$\vec{v}_{w/g} = 3\hat{i} \text{ m/s} \quad (\text{water relative to ground}) \\ \vec{v}_{b/w} = 5\hat{j} \text{ m/s} \quad (\text{boat relative to water})$$
2. Apply the relative velocity rule to find velocity relative to ground:

   $$\vec{v}_{b/g} = \vec{v}_{b/w} + \vec{v}_{w/g} = 3\hat{i} + 5\hat{j} \text{ m/s}$$
3. Calculate speed as the magnitude of the velocity vector:

   $$v = \sqrt{3^2 + 5^2} = \sqrt{34} \approx 5.83 \text{ m/s}$$
4. Time to cross depends only on the northward (across-river) velocity:

   $$t = \frac{\text{river width}}{v_y} = \frac{100}{5} = 20 \text{ s}$$
5. Calculate downstream distance from eastward velocity and crossing time:

   $$x = v_x t = 3(20) = 60 \text{ m}$$

> **Exam tip:** Always confirm your subscript order when setting up relative velocity problems. A quick check: the second subscript is the frame you are measuring velocity relative to, so your final answer's second subscript should match the frame the question asks for.

## Concept Check & Additional Worked Examples

**Check your understanding**

Test your understanding of 2D kinematics with these AP-style problems:

1. An object moves in the xy-plane with position given by $x(t) = 4t^2 - 2t$ and $y(t) = 1.5 \cos(2t)$, where position is in meters and t is in seconds. What is the magnitude of the object's acceleration at $t = 0$?

   - A) $5 \text{ m/s}^2$
   - B) $7.5 \text{ m/s}^2$
   - C) $10 \text{ m/s}^2$
   - D) $12.5 \text{ m/s}^2$

   *Why:* Acceleration is the second derivative of position, calculated component-wise: $a_x = 8 \text{ m/s}^2$, $a_y = -6 \text{ m/s}^2$ at $t=0$, so magnitude is $\sqrt{8^2 + (-6)^2} = 10 \text{ m/s}^2$.

**Worked example:** A projectile is launched from the edge of a 45 m high vertical cliff at an angle of $37^\circ$ above the horizontal, with an initial speed of $50 \text{ m/s}$. Ignore air resistance. (a) Find total time of flight. (b) Find horizontal impact distance. (c) Find impact speed. Use $g = 10 \text{ m/s}^2$.

1. (a) Set origin at launch, so $y_0 = 0$ and impact at $y = -45 \text{ m}$. Calculate initial components:

   $$v_{0x} = 50 \cos37^\circ = 40 \text{ m/s} \\ v_{0y} = 50 \sin37^\circ = 30 \text{ m/s}$$
2. Substitute into y-position equation and solve for t:

   $$-45 = 30t - 5t^2 \\ t^2 - 6t - 9 = 0$$
3. Take the positive root of the quadratic:

   $$t = 3 + 3\sqrt{2} \approx 7.24 \text{ s}$$
4. (b) Calculate horizontal distance:

   $$x = v_{0x} t = 40(7.24) \approx 290 \text{ m}$$
5. (c) Calculate velocity components and impact speed:

   $$v_x = 40 \text{ m/s}, v_y = 30 - 10(7.24) = -42.4 \text{ m/s} \\ v = \sqrt{40^2 + (-42.4)^2} \approx 58.3 \text{ m/s}$$

**Worked example:** A drone flies with velocity components $v_x(t) = 2t \text{ m/s}$ and $v_y(t) = (4 - t) \text{ m/s}$ for $0 \leq t \leq 4 \text{ s}$, starting from the origin at $t=0$. What is the straight-line distance from the launch point after 4 seconds?

1. Integrate each velocity component to get position, starting from $x(0)=0, y(0)=0$:
2. Integrate x-component:

   $$x(t) = \int_0^t 2\tau d\tau = t^2 \\ x(4) = 16 \text{ m}$$
3. Integrate y-component:

   $$y(t) = \int_0^t (4 - \tau) d\tau = 4t - 0.5t^2 \\ y(4) = 8 \text{ m}$$
4. Calculate straight-line distance from origin:

   $$d = \sqrt{16^2 + 8^2} = 8\sqrt{5} \approx 17.9 \text{ m}$$

## Common pitfalls

- **Wrong:** Using the range formula $R = v_0^2 \sin2\theta /g$ when launch height and landing height are different.
  - Why it fails: Students memorize the range formula for equal launch/landing height and automatically apply it to all projectile problems, even when launched from a cliff or roof.
  - Correct: Always start from the full y-position equation, solve for time of flight first, then calculate horizontal range from that time, regardless of height differences.
- **Wrong:** Adding x and y velocity components directly to get speed, instead of taking the Pythagorean magnitude.
  - Why it fails: Students forget velocity is a vector, and add scalar components like collinear 1D vectors.
  - Correct: For any speed calculation in 2D, always compute $v = \sqrt{v_x^2 + v_y^2}$ after finding components.
- **Wrong:** Using the wrong velocity component to calculate crossing time for relative motion river problems.
  - Why it fails: Students mix up which direction is across the river, and use the downstream velocity component for time calculation.
  - Correct: Align your coordinate system so that across the river is one axis, divide the river width by the velocity component along that axis to get time.
- **Wrong:** Forgetting to flip the sign of relative velocity when reversing subscript order.
  - Why it fails: Students only copy the magnitude of the velocity and ignore direction when switching reference frames.
  - Correct: Whenever you reverse subscript order for a relative velocity, immediately add a negative sign to the vector before further calculation.
- **Wrong:** Treating horizontal acceleration as non-zero in projectile motion when air resistance is ignored.
  - Why it fails: Students confuse the launch acceleration with acceleration after launch, or incorrectly add acceleration from air resistance when it is explicitly ignored.
  - Correct: For all standard projectile problems, explicitly write $a_x = 0$ at the start of your working.

## Cheatsheet

| Category | Formula | Notes |
| --- | --- | --- |
| General 2D Kinematic Vectors | $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}$ <br> $\vec{v}(t) = \frac{d\vec{r}}{dt} = \frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j}$ <br> $\vec{a}(t) = \frac{d\vec{v}}{dt} = \frac{d^2x}{dt^2}\hat{i} + \frac{d^2y}{dt^2}\hat{j}$ | Applies to all 2D motion, constant or non-constant acceleration |
| Constant Acceleration Kinematics | $\vec{v}(t) = \vec{v}_0 + \vec{a} t$ <br> $\vec{r}(t) = \vec{r}_0 + \vec{v}_0 t + \frac{1}{2}\vec{a} t^2$ | Only for constant $\vec{a}$; apply component-wise |
| Projectile Motion (Standard Coords) | $a_x = 0,\ a_y = -g$ <br> $x(t) = x_0 + v_0 \cos\theta \cdot t$ <br> $y(t) = y_0 + v_0 \sin\theta \cdot t - \frac{1}{2} g t^2$ | $+y$ is up; $g = 9.8 \text{ m/s}^2$; ignores air resistance |
| Projectile Maximum Height | $H = \frac{v_0^2 \sin^2\theta}{2g}$ | Only for maximum height measured from launch height, $v_y=0$ at peak |
| Projectile Time of Flight | $t = \frac{2 v_0 \sin\theta}{g}$ | Only when launch and landing height are equal |
| Projectile Range | $R = \frac{v_0^2 \sin 2\theta}{g}$ | Only when launch and landing height are equal; maximum at $\theta=45^\circ$ |
| Relative Velocity | $\vec{v}_{A/C} = \vec{v}_{A/B} + \vec{v}_{B/C}$ <br> $\vec{v}_{A/B} = -\vec{v}_{B/A}$ | Applies to inertial reference frames with constant relative velocity |

## What's next

Kinematics in two dimensions is the foundation for nearly all remaining topics in AP Physics C: Mechanics. You will next apply these vector decomposition and component-wise kinematic principles to Newton’s laws of motion in two dimensions, where you will solve for acceleration components and derive the resulting motion of objects under forces like gravity, tension, and friction. Without mastering the core techniques of this topic, solving problems involving inclined planes, circular motion, or variable-acceleration motion will be extremely difficult on exam day. This topic also feeds directly into energy and momentum conservation in 2D, where velocity components are required to solve collision problems.

- [Newton's Laws of Motion Overview](https://www.owlsprep.com/study/ap-physics-c-mech-u2-overview/)
- [Newton's First Law and Inertial Frames](https://www.owlsprep.com/study/ap-physics-c-mech-u2-newton-s-first-law-and/)
- [Newton's Second Law](https://www.owlsprep.com/study/ap-physics-c-mech-u2-newton-s-second-law/)

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