Uniform Linear Motion: Velocity

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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 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 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 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 $\vec{r}$. 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, 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.