Basic Physics

Motion in Two and Three Dimensions

by Admin•07/01/202531/01/2025

Motion in Two and Three Dimensions

Moving beyond a single plane, we will examine the motion of objects in both a plane and in space. In this section, we explore the fundamental concepts of motion in two and three dimensions, delving into the dynamics of vector quantities such as position, velocity, and acceleration.

2D/3D Motion – Clean Layout

Position Vector

The position vector describes an object’s location in space using coordinate components:

2D: $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}$
3D: $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}$

Example (Drone):
$\vec{r}(t) = 2t\hat{i} + 3t^2\hat{j} + 5\hat{k}$

Displacement

The vector difference between initial and final positions:

$\Delta \vec{r} = \vec{r}_f - \vec{r}_i = (x_2-x_1)\hat{i} + (y_2-y_1)\hat{j}$

Path independent
Includes direction information
Calculated through vector subtraction

Velocity

Rate of change of position with time:

Instantaneous: $\vec{v}(t) = \frac{d\vec{r}}{dt}$
Average: $\vec{v}_{\text{avg}} = \frac{\Delta \vec{r}}{\Delta t}$

Case Study:
For $\vec{r}(t) = t^2\hat{i} + 3t\hat{j}$ :
$\vec{v}(t) = 2t\hat{i} + 3\hat{j}$

Acceleration

Rate of change of velocity with time:

$\vec{a}(t) = \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2}$

Positive acceleration ≠ speeding up
Negative acceleration ≠ slowing down
Direction matters in vector context

Source

Serway, R. A., & Jewett, J. W. (2007). Fizik: Bilim insanları ve mühendisler için (5. baskı, Çev. F. Şahin & H. Çelik). Palme Yayıncılık

Kaya, S. (Yıl). Fizik 1: 1. Bölüm Notları. Gümüşhane Üniversitesi

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