2D Kinematics and Relative Motion: Study Notes for College Physics

2D Kinematics: Motion in Two Dimensions

Introduction to 2D Motion

Motion in two dimensions (2D) involves analyzing movement along both the x- and y-axes. This is commonly encountered in projectile motion, navigation, and other physical scenarios. 2D motion can be broken down into two independent 1D motions along perpendicular axes.

• Vector Equations: Used to describe position, displacement, velocity, and acceleration in 2D.

• Decomposition: Always break 2D motion into x and y components for analysis.

Position and Displacement Vectors

Position and displacement in 2D are represented as vectors, allowing calculation of magnitude and direction using vector equations.

• Position Vector: Specifies the location of a point relative to the origin.

• Displacement Vector: Represents the shortest path between two points.

Key Equations:

• Magnitude: $|\vec{r}| = \sqrt{x^2 + y^2}$

• Direction: $\theta = \tan^{-1}\left(\frac{y}{x}\right)$

• Components: $x = r \cos \theta$, $y = r \sin \theta$

Example: If a hiker moves 36m north of east, calculate the x and y components using the above equations.

Speed and Velocity in 2D

Speed and velocity measure how fast and in what direction an object moves between two points. Speed is a scalar (magnitude only), while velocity is a vector (magnitude and direction).

• Average Speed: $s = \frac{\text{distance}}{\text{time}}$

• Average Velocity: $|\vec{v}_{\text{avg}}| = \frac{|\vec{\Delta r}|}{\Delta t}$

• Direction: $\theta_v = \tan^{-1}\left(\frac{v_y}{v_x}\right)$

Example: You walk 40m in the x-direction and 30m in the y-direction in 10s. Calculate average speed and velocity.

Velocity Components and Vector Equations

Velocity in 2D has both x and y components. These can be calculated from displacement and time, or from magnitude and direction.

• Components from Displacement and Time:

  • $v_x = \frac{\Delta x}{\Delta t}$

  • $v_y = \frac{\Delta y}{\Delta t}$

• Components from Magnitude and Direction:

  • $v_x = v \cos \theta$

  • $v_y = v \sin \theta$

Example: A ball moves 10m/s at 53.1° above the x-axis. Find $v_x$ and $v_y$.

Acceleration in 2D

Acceleration causes a change in velocity (magnitude and/or direction). Like velocity, acceleration in 2D is analyzed using its components.

• Magnitude: $|\vec{a}| = \sqrt{a_x^2 + a_y^2}$

• Direction: $\theta_a = \tan^{-1}\left(\frac{a_y}{a_x}\right)$

• Components:

  • $a_x = a \cos \theta$

  • $a_y = a \sin \theta$

Example: A car accelerates at 3m/s² in a direction 37° northeast. Find $a_x$ and $a_y$.

Equations of Motion (UAM) in 2D

Uniformly Accelerated Motion (UAM) equations apply separately to x and y components in 2D motion.

Equation	X Component	Y Component
Final velocity	$v_{fx} = v_{ix} + a_x t$	$v_{fy} = v_{iy} + a_y t$
Displacement	$\Delta x = v_{ix} t + \frac{1}{2} a_x t^2$	$\Delta y = v_{iy} t + \frac{1}{2} a_y t^2$
Velocity squared	$v_{fx}^2 = v_{ix}^2 + 2 a_x \Delta x$	$v_{fy}^2 = v_{iy}^2 + 2 a_y \Delta y$

Example: A hockey puck slides at 8m/s east, then is accelerated at 3m/s² northeast for 5s. Find displacement.

Relative Motion and Reference Frames

Frames of Reference

All velocity measurements are made relative to a chosen frame of reference, typically the ground or Earth unless otherwise stated.

• Frame of Reference: The perspective from which motion is measured.

• Relative velocity depends on the observer's frame.

Example: On a moving walkway, the velocities of people walking are measured relative to the ground and to each other.

Relative Velocity Equation

Relative velocity is calculated by adding or subtracting velocities measured in different reference frames.

• General Equation: $v_{A/C} = v_{A/B} + v_{B/C}$

• Subscripts indicate the object and reference frame.

• When flipping subscripts, the sign of the velocity reverses: $v_{A/B} = -v_{B/A}$

Example: If a car moves at 45m/s east relative to the ground, and a truck moves at 60m/s east, the velocity of the truck relative to the car is $v_{T/C} = v_{T/G} - v_{C/G}$.

Solving Relative Velocity Problems

• Draw diagrams and identify all objects and reference frames.

• Write given velocities with correct subscript notation.

• Apply the relative velocity equation according to the rules.

• Solve for the desired quantity.

Step	Description
1	Draw diagram, identify objects & references
2	Write given velocities with subscript notation
3	Write relative velocity equation according to rules
4	Solve

Example: A boat travels in a river flowing at 4m/s. If the boat makes the trip in 250s, calculate the speed of the boat relative to the water.

Summary Table: Key Equations in 2D Kinematics

Quantity	Equation
Position/Displacement Magnitude	$|\vec{r}| = \sqrt{x^2 + y^2}$
Position/Displacement Direction	$\theta = \tan^{-1}\left(\frac{y}{x}\right)$
Velocity Magnitude	$|\vec{v}| = \sqrt{v_x^2 + v_y^2}$
Velocity Direction	$\theta_v = \tan^{-1}\left(\frac{v_y}{v_x}\right)$
Acceleration Magnitude	$|\vec{a}| = \sqrt{a_x^2 + a_y^2}$
Acceleration Direction	$\theta_a = \tan^{-1}\left(\frac{a_y}{a_x}\right)$

Additional info: These notes expand on the original brief points and examples, providing full academic context, definitions, and formulas for 2D kinematics and relative motion, suitable for college-level physics students.