Kinematics in Two Dimensions

Overview

Motion in two dimensions involves analyzing motion in a plane using vector quantities. The key insight is that horizontal and vertical motions are independent.

Position and Displacement Vectors

Position Vector

$\vec{r} = x\hat{i} + y\hat{j}$

Displacement Vector

$\Delta\vec{r} = \vec{r}_2 - \vec{r}_1 = \Delta x\hat{i} + \Delta y\hat{j}$

Magnitude of Displacement

$|\Delta\vec{r}| = \sqrt{(\Delta x)^2 + (\Delta y)^2}$

Velocity Vector

Average Velocity

$\vec{v}_{\text{avg}} = \frac{\Delta\vec{r}}{\Delta t} = \frac{\Delta x}{\Delta t}\hat{i} + \frac{\Delta y}{\Delta t}\hat{j}$

Components

$v_x = \frac{dx}{dt}, \quad v_y = \frac{dy}{dt}$

Magnitude and Direction

$|\vec{v}| = \sqrt{v_x^2 + v_y^2}$

$\theta = \tan^{-1}\left(\frac{v_y}{v_x}\right)$

Acceleration Vector

$\vec{a} = \frac{dv_x}{dt}\hat{i} + \frac{dv_y}{dt}\hat{j} = a_x\hat{i} + a_y\hat{j}$

Projectile Motion

An object launched into the air with only gravity acting on it.

Assumptions

• Air resistance is negligible
• $g$ is constant (9.8 m/s² downward)
• Earth's rotation is ignored

Initial Conditions

$v_{0x} = v_0 \cos\theta$

$v_{0y} = v_0 \sin\theta$

Equations of Motion

Horizontal (no acceleration):

$x = x_0 + v_{0x}t$

$v_x = v_{0x} = \text{constant}$

Vertical (constant acceleration = $-g$):

$y = y_0 + v_{0y}t - \frac{1}{2}gt^2$

$v_y = v_{0y} - gt$

$v_y^2 = v_{0y}^2 - 2g(y - y_0)$

Key Results

Time of flight (landing at same height):

$T = \frac{2v_0 \sin\theta}{g}$

Maximum height:

$H = \frac{v_0^2 \sin^2\theta}{2g}$

Range (horizontal distance):

$R = \frac{v_0^2 \sin(2\theta)}{g}$

Maximum range occurs at $\theta = 45°$

Trajectory Equation

$y = x\tan\theta - \frac{gx^2}{2v_0^2 \cos^2\theta}$

Relative Velocity

In One Dimension

$v_{A/B} = v_A - v_B$

Velocity of A relative to B

In Two Dimensions

$\vec{v}_{A/C} = \vec{v}_{A/B} + \vec{v}_{B/C}$

Example: River Crossing

A boat in a river experiences:

• Its velocity relative to water: $\vec{v}_{B/W}$
• Water's velocity relative to ground: $\vec{v}_{W/G}$
• Boat's velocity relative to ground: $\vec{v}_{B/G} = \vec{v}_{B/W} + \vec{v}_{W/G}$

Uniform Circular Motion

Object moving in a circle at constant speed has centripetal acceleration:

$a_c = \frac{v^2}{r} = \omega^2 r$

Direction: Always toward the center of the circle

Period and Frequency

$T = \frac{2\pi r}{v} = \frac{2\pi}{\omega}$

$f = \frac{1}{T} = \frac{v}{2\pi r}$

Examples

Example 1: Projectile from Ground

A ball is kicked at 20 m/s at 30° above horizontal. Find range and max height.

$v_{0x} = 20\cos(30°) = 17.3 \text{ m/s}$

$v_{0y} = 20\sin(30°) = 10 \text{ m/s}$

$H = \frac{v_{0y}^2}{2g} = \frac{100}{2 \times 9.8} = 5.1 \text{ m}$

$R = \frac{v_0^2 \sin(60°)}{g} = \frac{400(0.866)}{9.8} = 35.3 \text{ m}$

Example 2: Projectile from Height

A ball is thrown horizontally at 15 m/s from a 20 m cliff.

Time to fall:

$t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{40}{9.8}} = 2.02 \text{ s}$

Horizontal distance:

$x = v_{0x}t = 15 \times 2.02 = 30.3 \text{ m}$