# Uniform Linear Motion: Velocity
Source: algebrica.org - CC BY-NC 4.0
https://algebrica.org/velocity/
Fetched from algebrica.org post 14715; source modified 2026-03-20T22:29:55.
## Introduction
Kinematics is the study of the motion of objects, describing their position, velocity, and [acceleration](../acceleration.md) over time. Before diving into a detailed explanation of these phenomena, it is essential to introduce some key concepts.
- A material point is an idealized object whose size is considered negligible relative to the distances involved in its motion.
- The trajectory is the path traced by a material point as it moves through space.
- A motion is said to be rectilinear if its trajectory lies along a straight line.
If a material point moves along a straight-line path at constant velocity, meaning that the distances traveled are proportional to the time intervals taken, the motion is called **uniform rectilinear motion.**
## Velocity
Let’s consider two points, \\( x\_1 \\) and \\( x\_2 \\), representing the position \\( P \\) of a point at two successive moments in time, \\( t\_1 \\) and \\( t\_2 \\), respectively.
We can express the following relationship:
\\[
\\frac{x\_2 - x\_1}{t\_2 - t\_1} = v
\\]
This relationship shows that the distance traveled is proportional to the elapsed time, and the ratio remains constant, equal to the magnitude \\( v \\). The **instantaneous scalar speed** is defined as the limit of the ratio as the time [interval](../intervals.md) approaches zero:
\\[
v\_s = \\lim\_{\\Delta t \\to 0} \\frac{\\Delta x}{\\Delta t} = \\lim\_{\\Delta t \\to 0} \\frac{x(t + \\Delta t) - x(t)}{\\Delta t}
\\]
where \\( v \\) represents the instantaneous speed, \\( \\Delta x \\) is the displacement, and \\( \\Delta t \\) is the time interval. Therefore, the instantaneous scalar speed is given by the [derivative](../derivatives.md) of \\(x = x(t)\\) with respect to time.
* * *
Let us now imagine that the position \\( P \\) of the point at time \\( t \\) is identified by the displacement [vector](../vectors.md) \\( \\vec{r} \\). The displacement from \\( P \\) to \\( P’ \\) occurs over a time interval \\( \\Delta t \\) and is represented by the vector \\( \\Delta \\vec{r} \\). We have:
\\[
\\lim\_{\\Delta t \\to 0} \\frac{\\Delta \\mathbf{r}}{\\Delta t} = \\frac{d\\mathbf{r}}{dt} = \\mathbf{v}
\\]
Thus, we can define the **velocity vector** as:
\\[
\\mathbf{v} = \\frac{dx}{dt} = \\mathbf{i} \\frac{dx(t)}{dt}
\\]
where \\(\\mathbf{i}\\) represents a directed and oriented vector. The velocity vector is tangent to the trajectory at each point, oriented according to the direction of motion, and has a magnitude equal to the scalar speed.
- In uniform rectilinear motion, the velocity vector remains constant.
- In uniform rectilinear motion, the trajectory’s position-time equation is a [straight line](../lines.md), meaning the position varies linearly with time.
* * *
Velocity is measured in units of length multiplied by time raised to the power of \\(-1\\), and its standard unit is meters per second \\((\\text{ms}^{-1})\\).
## Example
Imagine a car traveling along a straight road at a constant speed of \\( v = 20\\ \\mathrm{m/s} \\). Since the velocity is constant, the distance traveled by the car is directly proportional to the elapsed time. The position \\( x(t) \\) of the car at any time \\( t \\) can be expressed as:
\\[
x(t) = x\_0 + v t
\\]
where:
- \\( x\_0 \\) is the initial position (at \\( t = 0 \\)).
- \\( v \\) is the constant velocity.
- \\( t \\) is the time elapsed.
This means that for every second that passes, the car moves exactly 20 meters forward, without speeding up or slowing down. Let’s summarize the data in a table:
\\[
\\begin{array}{c|c} \\text{Time } (\\mathrm{s}) & \\text{Position } (\\mathrm{m}) \\\\ \\hline 0 & 0 \\\\ 1 & 20 \\\\ 2 & 40 \\\\ 3 & 60 \\\\ 4 & 80 \\\\ \\vdots & \\vdots \\end{array}
\\]
## Glossary
- Kinematics: the study of the motion of objects, describing their position, velocity, and acceleration over time.
- Material point: an idealized object whose size is considered negligible relative to the distances involved in its motion.
- Trajectory: the path traced by a material point as it moves through space.
- Rectilinear motion: motion whose trajectory lies along a straight line.
- Uniform rectilinear motion: motion along a straight line with constant velocity, where distances traveled are proportional to time intervals.
- Scalar speed: the limit of the ratio of displacement to time interval as the time interval approaches zero; the magnitude of the velocity vector.
- Velocity vector: the rate of change of the displacement vector with respect to time; a vector tangent to the trajectory, oriented in the direction of motion, with a magnitude equal to the scalar speed.
## What is velocity
- Velocity describes how fast and in what direction an object moves.
- Scalar velocity refers to the absolute value of velocity, representing only the speed of the object without considering the direction.
- Vector velocity is the rate of change of position with respect to time, expressed as a vector tangent to the trajectory and oriented in the direction of motion.
