Kinematics in Two Dimensions: Position, Displacement, and Velocity

Kinematics in Two Dimensions

Introduction

Kinematics in two dimensions involves analyzing the motion of objects that move along curved paths in a plane. This topic is fundamental in physics, as it extends the concepts of position, displacement, and velocity from one dimension to two, allowing for a more complete description of real-world motion.

Position in Two Dimensions

Position Vector

The position of an object in two dimensions is described by a vector that points from the origin to the object's location in the plane.

• Position vector (r): An arrow from the origin to the object's position.

• Components: x (horizontal) and y (vertical).

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

• The position vector provides a 'picture' of the object's location at any instant.

Example:

If a particle is at (x, y) = (3 m, 4 m), its position vector is $\vec{r} = 3\hat{i} + 4\hat{j}$ m.

Displacement and Velocity in Two Dimensions

Displacement Vector

Displacement is the change in position of an object, represented by a vector pointing from the initial to the final position.

• Given two position vectors $\vec{r}_1$ and $\vec{r}_2$ at times $t_1$ and $t_2$:

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

Average Velocity

Average velocity is the rate of change of displacement over time.

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

• The direction of average velocity is the same as the displacement vector $\Delta \vec{r}$.

Example:

If a particle moves from (1 m, 2 m) to (4 m, 6 m) in 2 seconds:

• $\Delta x = 4 - 1 = 3$ m

• $\Delta y = 6 - 2 = 4$ m

• $\vec{v}_{avg} = \frac{3}{2}\hat{i} + \frac{4}{2}\hat{j} = 1.5\hat{i} + 2\hat{j}$ m/s

Instantaneous Velocity in Two Dimensions

Definition and Calculation

Instantaneous velocity is the velocity of an object at a specific instant, found by taking the derivative of the position vector with respect to time.

• Formula: $\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{r}}{\Delta t} = \frac{d\vec{r}}{dt} = \frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j}$

• Components: $v_x = \frac{dx}{dt}$ and $v_y = \frac{dy}{dt}$

• The instantaneous velocity vector is tangent to the trajectory at each point.

Example:

If $x(t) = t^2$ and $y(t) = 3t$, then:

• $v_x = \frac{d}{dt}(t^2) = 2t$

• $v_y = \frac{d}{dt}(3t) = 3$

• $\vec{v} = 2t\hat{i} + 3\hat{j}$

Velocity Components and Direction

The velocity vector can be expressed in terms of its magnitude and direction, often using the angle $\theta$ measured counterclockwise from the +x axis.

• Given speed $v$ and angle $\theta$:

• $v_x = v \cos \theta$

• $v_y = v \sin \theta$

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

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

Example:

If $v = 10$ m/s and $\theta = 30^\circ$:

• $v_x = 10 \cos 30^\circ = 8.66$ m/s

• $v_y = 10 \sin 30^\circ = 5.0$ m/s

Summary Table: Position, Displacement, and Velocity in 2D

Quantity	Vector Form	Components
Position	$\vec{r}$	$x, y$
Displacement	$\Delta \vec{r}$	$\Delta x, \Delta y$
Average Velocity	$\vec{v}_{avg}$	$\frac{\Delta x}{\Delta t}, \frac{\Delta y}{\Delta t}$
Instantaneous Velocity	$\vec{v}$	$\frac{dx}{dt}, \frac{dy}{dt}$

Additional info: These notes expand on the brief points in the slides, providing full definitions, formulas, and examples for each concept. The summary table is inferred for clarity and completeness.