Motion in Two and Three Dimensions: Vectors, Velocity, and Acceleration

Motion in Two and Three Dimensions

Introduction

This chapter explores the representation and analysis of motion in two and three dimensions using vectors. Students will learn how to describe position, velocity, and acceleration as vectors, interpret their components, and apply these concepts to projectile and circular motion.

Position Vectors

The position vector locates a particle in space relative to an origin. In two or three dimensions, it is expressed in terms of its components along the coordinate axes.

• Definition: The position vector r from the origin to point P is given by:

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

• Components: x, y, and z are the coordinates of the particle.

• Example: If a particle is at (2, -4, 8), then $\vec{r} = 2\hat{i} - 4\hat{j} + 8\hat{k}$.

Displacement Vector

The displacement vector represents the change in position of a particle between two points.

• Definition: The displacement from $\vec{r}_1$ to $\vec{r}_2$ is:

$\Delta \vec{r} = \vec{r}_2 - \vec{r}_1 = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}$

• Direction: Points from the initial to the final position.

• Example: If $\vec{r}_1 = 2\hat{i} - 4\hat{j} + 8\hat{k}$ and $\vec{r}_2 = 6\hat{i} + 0\hat{j} + 10\hat{k}$, then $\Delta \vec{r} = 4\hat{i} + 4\hat{j} + 2\hat{k}$.

Average Velocity

Average velocity is defined as the displacement divided by the time interval over which the displacement occurs.

• Formula:

$\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}$

• Direction: Same as the displacement vector.

• Units: meters per second (m/s).

• Example: If $\Delta \vec{r} = 4\hat{i} + 4\hat{j} + 2\hat{k}$ and $\Delta t = 2$ s, then $\vec{v}_{avg} = 2\hat{i} + 2\hat{j} + 1\hat{k}$ m/s.

Instantaneous Velocity

Instantaneous velocity is the rate of change of position with respect to time at a specific instant.

• Formula:

$\vec{v} = \frac{d\vec{r}}{dt}$

• Components: $v_x = \frac{dx}{dt}$, $v_y = \frac{dy}{dt}$, $v_z = \frac{dz}{dt}$

• Magnitude: $|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$

• Direction: Tangent to the particle's path.

Unit Vectors

Unit vectors are vectors of magnitude 1 that indicate direction along coordinate axes.

• Notation: $\hat{i}$ (x-axis), $\hat{j}$ (y-axis), $\hat{k}$ (z-axis)

• Any vector can be written as a sum of its components times the unit vectors.

• Example: $\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$

Acceleration

Acceleration is the rate of change of velocity with respect to time. It can be average or instantaneous.

• Average acceleration:

$\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t}$

• Instantaneous acceleration:

$\vec{a} = \frac{d\vec{v}}{dt}$

• Components: $a_x = \frac{dv_x}{dt}$, $a_y = \frac{dv_y}{dt}$, $a_z = \frac{dv_z}{dt}$

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

• Direction: Can be parallel or perpendicular to the path; parallel component changes speed, perpendicular changes direction.

Projectile Motion

Projectile motion describes the motion of an object launched into the air, subject only to gravity (neglecting air resistance).

• Trajectory: The path followed is a parabola.

• Equations of motion:

$\begin{align*} x(t) &= x_0 + v_{x,0} t \\ y(t) &= y_0 + v_{y,0} t - \frac{1}{2} g t^2 \end{align*}$

• Horizontal and vertical motions are independent.

• Example: A ball thrown with initial velocity $v_0$ at angle $\theta$:

$\begin{align*} v_{x,0} &= v_0 \cos \theta \\ v_{y,0} &= v_0 \sin \theta \end{align*}$

• Maximum height and range:

$H_{max} = \frac{v_0^2 \sin^2 \theta}{2g}$ $R = \frac{v_0^2 \sin 2\theta}{g}$

• Applications: Sports (baseball, basketball), ballistics.

Relative Velocity

Relative velocity describes how the velocity of an object appears from different frames of reference.

• Formula:

$\vec{v}_{A/B} = \vec{v}_A - \vec{v}_B$

• Example: If a train moves at 20 m/s east and a passenger walks at 2 m/s east inside the train, the passenger's velocity relative to the ground is 22 m/s east.

Summary Table: Key Vector Quantities

Quantity	Definition	Formula	Direction
Position Vector	Location of particle from origin	$\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$	From origin to particle
Displacement	Change in position	$\Delta \vec{r} = \vec{r}_2 - \vec{r}_1$	From initial to final position
Average Velocity	Displacement per unit time	$\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}$	Same as displacement
Instantaneous Velocity	Rate of change of position	$\vec{v} = \frac{d\vec{r}}{dt}$	Tangent to path
Acceleration	Rate of change of velocity	$\vec{a} = \frac{d\vec{v}}{dt}$	Parallel/perpendicular to path

Additional info:

• Unit vectors are essential for expressing vectors in component form and for vector addition/subtraction.

• Projectile motion analysis assumes negligible air resistance and a flat Earth for introductory problems.

• Relative velocity is crucial in navigation, transport, and physics problems involving multiple moving frames.