KAHIBARO
Table of Contents
Direction and Rate of Motion
In two and three dimensional motion, position is described by a position vector. The velocity vector tells us how that position changes with time. It gives both how fast an object moves and the direction in which it moves.
If the position vector is written as $\vec r(t)$, then the velocity vector is defined by
$$
\vec v(t) = \frac{d\vec r}{dt}
$$
This means velocity is the time derivative of position. In words, velocity points in the direction the object is moving at that instant.
The velocity vector is the rate of change of position:
$$
\vec v = \frac{d\vec r}{dt}
$$
Its direction matters, not just its size.
Component Form
In Cartesian coordinates, the position vector in two dimensions can be written as
$$
\vec r(t) = x(t)\hat{i} + y(t)\hat{j}
$$
Then the velocity vector is
$$
\vec v(t) = \frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j}
$$
So we define the velocity components as
$$
v_x = \frac{dx}{dt}, \qquad v_y = \frac{dy}{dt}
$$
In three dimensions,
$$
\vec r(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}
$$
and
$$
\vec v(t) = \frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j} + \frac{dz}{dt}\hat{k}
$$
so
$$
v_x = \frac{dx}{dt}, \qquad v_y = \frac{dy}{dt}, \qquad v_z = \frac{dz}{dt}
$$
Each component tells how quickly the object moves along one coordinate axis.
Velocity as Displacement per Time Interval
Over a finite time interval $\Delta t$, the average velocity vector is
$$
\vec v_{\text{avg}} = \frac{\Delta \vec r}{\Delta t}
$$
where $\Delta \vec r$ is the displacement vector. As the time interval becomes very small, the average velocity approaches the instantaneous velocity vector.
This shows an important idea. In multiple dimensions, velocity is connected to displacement, not total distance traveled. An object can move along a curved path, but at each instant its velocity still points in one specific direction.
Average velocity uses displacement:
$$
\vec v_{\text{avg}} = \frac{\Delta \vec r}{\Delta t}
$$
Instantaneous velocity is the limit of this as $\Delta t \to 0$.
Magnitude of the Velocity Vector
The magnitude of the velocity vector is the speed. In two dimensions,
$$
|\vec v| = \sqrt{v_x^2 + v_y^2}
$$
In three dimensions,
$$
|\vec v| = \sqrt{v_x^2 + v_y^2 + v_z^2}
$$
So speed is a scalar, while velocity is a vector.
| Quantity | Type | Meaning | ||
|---|---|---|---|---|
| $\vec v$ | Vector | Speed and direction of motion | ||
| $ | \vec v | $ | Scalar | Speed only |
| $v_x, v_y, v_z$ | Scalars | Velocity along each axis |
Geometric Meaning
At any point on a path, the velocity vector is tangent to the path. This is true whether the object moves in a straight line or along a curve.
If the path bends, the direction of the velocity vector changes from point to point. Even if the speed stays constant, the velocity vector can still change because its direction changes.
Interpreting the Components
The sign of each velocity component tells the direction along that axis.
If $v_x > 0$, the object moves in the positive $x$ direction. If $v_x < 0$, it moves in the negative $x$ direction. The same idea applies to $v_y$ and $v_z$.
An object can have one component equal to zero while still moving. For example, if $v_y = 0$ but $v_x \neq 0$, the motion is purely horizontal.
Finding Velocity from Position Functions
Suppose the position is
$$
x(t) = 3t, \qquad y(t) = 4t^2
$$
Then
$$
v_x = \frac{dx}{dt} = 3, \qquad v_y = \frac{dy}{dt} = 8t
$$
So the velocity vector is
$$
\vec v(t) = 3\hat{i} + 8t\hat{j}
$$
Its magnitude is
$$
|\vec v| = \sqrt{3^2 + (8t)^2} = \sqrt{9 + 64t^2}
$$
This example shows that the direction and size of the velocity vector may both change with time.
Direction Angle of Velocity in Two Dimensions
In two dimensional motion, the direction of the velocity vector can be described by an angle $\theta$ measured from the positive $x$ axis. If $v_x$ and $v_y$ are known, then
$$
\tan\theta = \frac{v_y}{v_x}
$$
Care is needed with signs, because the signs of $v_x$ and $v_y$ determine the correct quadrant.
For two dimensional motion,
$$
|\vec v| = \sqrt{v_x^2 + v_y^2}
$$
and, when $v_x \neq 0$,
$$
\tan\theta = \frac{v_y}{v_x}
$$
These describe the size and direction of the velocity vector.
A Visual Example
In this picture, the velocity vector is built from horizontal and vertical components. The vector points in the actual direction of motion.
Key Idea
Velocity vectors are essential in two and three dimensional motion because motion is no longer described by a sign alone. We must know both the amount of motion per unit time and the direction in space.
In more than one dimension, velocity is not completely described by a single number. You need its components or its magnitude and direction.