# Uniformly Accelerated Motion: Acceleration
Source: algebrica.org - CC BY-NC 4.0
https://algebrica.org/acceleration/
Fetched from algebrica.org post 14802; source modified 2026-03-20T22:29:24.
## Introduction
Kinematics is the study of the motion of objects, describing their position, [velocity](../velocity.md), 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](../lines.md) path under constant acceleration, meaning that the rate of change of velocity remains uniform over time, the motion is called **uniformly accelerated rectilinear motion**.
## Acceleration
Let us consider a particle moving along a straight-line trajectory, where the position as a function of time is not described by a [linear equation](../linear-equations.md). Let \\( P\_1 \\) and \\( P\_2 \\) denote two positions of the material point along the \\( x \\)-axis at times \\( t\_1 \\) and \\( t\_2 \\), respectively. We denote by \\( \\mathbf{v}\_1 \\) and \\( \\mathbf{v}\_2 \\) the corresponding velocity [vectors](../vectors.md), with \\( \\mathbf{v}\_1 \\neq \\mathbf{v}\_2 \\). The **vector acceleration** is defined as the following limit:
\\[
\\mathbf{a} = \\lim\_{\\Delta t \\to 0} \\frac{\\Delta \\mathbf{v}}{\\Delta t} = \\frac{d\\mathbf{v}}{dt}
\\]
We have seen, by analyzing the [velocity](../velocity.md), that:
\\[
\\lim\_{\\Delta t \\to 0} \\frac{\\Delta \\mathbf{r}}{\\Delta t} = \\frac{d\\mathbf{r}}{dt} = \\mathbf{v}
\\]
Thus, we have:
\\[
\\mathbf{a} = \\frac{d}{dt}\\left( \\frac{d\\mathbf{r}}{dt} \\right) = \\frac{d^2 \\mathbf{r}}{dt^2}
\\]
* * *
Starting from the general expression of acceleration it is possible to introduce the concept of **tangential acceleration** As a point \\( P \\) travels along a given path, the acceleration vector \\( \\mathbf{a} \\) can be broken down into two components:
- One tangential to the trajectory.
- One normal to the trajectory (also called centripetal acceleration that points toward the center of the curvature of the path).
The tangential acceleration, denoted by \\( \\mathbf{a}\_t \\), corresponds to the variation of the speed over time. It is defined as:
\\[
a\_t = \\frac{dv}{dt} = \\mathbf{i} \\, a\_t
\\]
where \\( v \\) represents the magnitude of the velocity vector \\( \\mathbf{v} \\) and \\(\\mathbf{i}\\) represents a directed and oriented vector.
- If the magnitude of the velocity changes, there is tangential acceleration \\((a\_t \\neq 0)\\).
- If the magnitude of the velocity remains constant, the tangential acceleration is zero \\((a\_t = 0)\\).
* * *
Uniformly accelerated motion is a type of motion in which the tangential acceleration \\( a\_t \\) is constant at every point and equal to the average acceleration over any time [interval](../intervals.md). We have:
\\[
\\frac{v - v\_0}{t-t\_0} = a\_t
\\]
Starting from this formula, solving for \\( v \\) and assuming \\( t\_0 = 0 \\), we obtain:
\\[
v = v\_0 + a\_t t
\\]
In this way, derived the expression for velocity based on the definition of acceleration. Starting from the expression of velocity as a function of time we can derive the equation of motion by [integrating](../indefinite-integrals.md) with respect to time:
\\[
y = \\int\_0^t v(t) \\, dt = \\int\_0^t (v\_0 + a\_t t) \\, dt
\\]
Evaluating the integral, we obtain:
\\[
y = v\_0 t + \\frac{1}{2} a\_t t^2
\\]
where \\( y \\) represents the displacement of the material point along the trajectory as a function of time.
