Motion in Two and Three Dimensions

 

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
7 Jan, 2025