Kinematics in Three Dimensions: Position, Velocity, and Instantaneous Velocity

Kinematics in Three Dimensions

Position Vector in 3D Space

In three-dimensional kinematics, the position vector describes the location of a particle in space using three coordinates (x, y, z). This vector is fundamental for analyzing motion in physics.

• Position Vector: $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, where $x$, $y$, and $z$ are the coordinates along the respective axes, and $\hat{i}$, $\hat{j}$, $\hat{k}$ are the unit vectors in the x, y, and z directions.

• Example: If a particle is at (2, 3, 5), its position vector is $2\hat{i} + 3\hat{j} + 5\hat{k}$.

Average Velocity Vector

The average velocity of a particle is defined as the change in position vector divided by the change in time. It provides a measure of the overall rate of displacement over a time interval.

• Formula: $\vec{v}_{\text{ave}} = \frac{\Delta \vec{r}}{\Delta t} = \frac{\vec{r}_2 - \vec{r}_1}{t_2 - t_1}$

• Where:

  • $\vec{r}_1$ = Position at time $t_1$

  • $\vec{r}_2$ = Position at time $t_2$

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

• Interpretation: The average velocity vector points from the initial to the final position and its magnitude gives the speed over the interval.

• Example: If a particle moves from (1, 2, 3) at $t_1 = 0$ s to (4, 6, 5) at $t_2 = 2$ s, then $\vec{v}_{\text{ave}} = \frac{(4-1)\hat{i} + (6-2)\hat{j} + (5-3)\hat{k}}{2-0} = \frac{3\hat{i} + 4\hat{j} + 2\hat{k}}{2} = 1.5\hat{i} + 2\hat{j} + 1\hat{k}$ m/s.

Instantaneous Velocity

The instantaneous velocity is the rate of change of the position vector with respect to time. It is a vector quantity and is always tangent to the particle's path at any given point.

• Definition: $\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{r}}{\Delta t} = \frac{d\vec{r}}{dt}$

• Component Form: $\frac{d\vec{r}}{dt} = \frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j} + \frac{dz}{dt}\hat{k}$

• Each component:

  • $v_x = \frac{dx}{dt}$

  • $v_y = \frac{dy}{dt}$

  • $v_z = \frac{dz}{dt}$

• Interpretation: At every point along the path, the instantaneous velocity vector is tangent to the trajectory.

• Example: If $x(t) = t^2$, $y(t) = 2t$, $z(t) = 3$, then $\vec{v} = 2t\hat{i} + 2\hat{j} + 0\hat{k}$.

Summary Table: Position and Velocity in 3D

Quantity	Vector Form	Component Form	Physical Meaning
Position	$\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$	x, y, z	Location in space
Average Velocity	$\vec{v}_{\text{ave}} = \frac{\Delta \vec{r}}{\Delta t}$	$\frac{\Delta x}{\Delta t}$, $\frac{\Delta y}{\Delta t}$, $\frac{\Delta z}{\Delta t}$	Rate of change of position over interval
Instantaneous Velocity	$\vec{v} = \frac{d\vec{r}}{dt}$	$\frac{dx}{dt}$, $\frac{dy}{dt}$, $\frac{dz}{dt}$	Rate of change of position at a point

Additional info: The notes emphasize that velocity is always tangent to the path, which is a key concept in understanding motion in multiple dimensions. This is foundational for further study in kinematics, dynamics, and vector calculus in physics.