Uniformly Accelerated Rectilinear Motion (UARM). Equations

08/05/2026

The online uniformly accelerated rectilinear motion (UARM) simulations on this page teach you in a practical way what this motion is like and what are the main equations that describe it.

This Thematic Unit is part of our Physics collection

Constant Acceleration

Variation of an object’s velocity by equal amounts during equal time intervals.

Free Fall

Particular case of uniformly accelerated motion where the acceleration is gravity and there is no air resistance.

Initial Velocity

Velocity that a body possesses at the exact moment time begins to be counted (t=0).

Instantaneous Velocity

Velocity of a body at a specific moment in time or at a certain point in its trajectory.

Negative Acceleration (Deceleration)

Change in velocity where its value decreases over time, also known as braking.

Positive Acceleration

Change in velocity where its value increases over time in the direction of motion.

Velocity Variation

Difference between the final velocity and the initial velocity of a body during a time interval.

Velocity-Time Graph

Visual representation where the slope of the straight line indicates the value of the movement’s acceleration.

What is uniformly accelerated rectilinear motion (UARM)?

Uniformly accelerated rectilinear motion (UARM), also called uniformly varied rectilinear motion (UVRM), is a type of motion in which an object experiences a constant change in velocity over time. In this type of motion, the acceleration of the object remains constant, meaning that it experiences a uniform acceleration.

Uniformly accelerated rectilinear motion (UARM) equations

To describe this type of motion, several equations are used, the most important of which are the velocity equation and the distance equation.

Velocity equation in UARM

The equation of velocity in uniformly accelerated rectilinear motion is:

v = v₀ + at

In this equation, v represents the velocity at a given instant, v₀ is the initial velocity, a is the acceleration and t is the time elapsed since the initial instant. This equation allows us to calculate the final velocity of an object as a function of its initial velocity, acceleration and time.

Equation of the distance in UARM

Another important equation is the equation of distance traveled in uniformly accelerated motion:

d = v₀t + (1/2)at².

In this equation, d represents the distance traveled, v₀ is the initial velocity, t is the time elapsed since the initial instant and a is the acceleration. This equation allows us to calculate the distance traveled by the object as a function of its initial velocity, acceleration and time.

Other UARM equations

From these basic UARM equations, other equations can be derived and calculations can be performed to determine the velocity, position and other parameters of the moving object. Two examples are the equation of distance as a function of velocity and the equation of velocity as a function of distance.

Equation of distance as a function of velocity

This equation allows one to calculate the distance traveled using the initial and final velocities, along with the acceleration. Its expression is:

d = (v² – v₀²) / (2a)

Equation of velocity as a function of distance

This equation determines the final velocity as a function of initial position, distance traveled and acceleration. Its expression is:

v² = v₀² + 2a·d

These derived UARM equations complement the fundamental ones, facilitating the analysis of UARM in various practical applications.

UARM in everyday life

Uniformly accelerated motion is encountered in a variety of scenarios in everyday life and science. Examples include the launching of objects in projectiles, the motion of accelerating or decelerating vehicles, and the free fall of objects under the influence of gravity.

These online uniformly accelerated rectilinear motion (UARM) simulations are very useful for understanding this basic but important type of motion.

Constant Acceleration

Variation of an object’s velocity by equal amounts during equal time intervals.

Free Fall

Particular case of uniformly accelerated motion where the acceleration is gravity and there is no air resistance.

Initial Velocity

Velocity that a body possesses at the exact moment time begins to be counted (t=0).

Instantaneous Velocity

Velocity of a body at a specific moment in time or at a certain point in its trajectory.

Negative Acceleration (Deceleration)

Change in velocity where its value decreases over time, also known as braking.

Positive Acceleration

Change in velocity where its value increases over time in the direction of motion.

Velocity Variation

Difference between the final velocity and the initial velocity of a body during a time interval.

Velocity-Time Graph

Visual representation where the slope of the straight line indicates the value of the movement’s acceleration.

Explore the exciting STEM world with our free, online, simulations and accompanying companion courses! With them you’ll be able to experience and learn hands-on. Take this opportunity to immerse yourself in virtual experiences while advancing your education – awaken your scientific curiosity and discover all that the STEM world has to offer!

Uniformly accelerated rectilinear motion (UARM) simulations

UARM equations

Free fall

This simulation allows you to view graphs of distance and velocity as a function of time for an object in free fall.

Free motion

Giants of science

“If I have seen further, it is by standing on the shoulders of giants”

Isaac Newton

René Descartes

1596

–

1650

René Descartes developed analytic geometry and contributed to classical mechanics and the study of optics

“Pienso, luego existo”

Archimedes

287 a.C.

–

212 a.C.

Archimedes established fundamental principles of mechanics and hydrostatics, including the famous Archimedes’ principle of buoyancy in fluids

“Give me a place to stand, and I will move the world”

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Test your knowledge

What is uniformly accelerated rectilinear motion and how is it described mathematically?

Uniformly accelerated rectilinear motion (UARM) is a type of movement in which an object travels in a straight line while its acceleration remains constant. This means the object’s speed changes steadily over time. To describe this motion, equations relate velocity, acceleration, time, and distance. For instance, the velocity at a given moment is obtained by adding the initial velocity and the product of acceleration by time, while the distance traveled is calculated by adding the initial velocity times time plus half the product of acceleration by the square of the time. These equations allow us to predict the behavior of objects under constant acceleration.

What are the main equations of UARM and what does each one allow us to find?

The main equations of UARM let us calculate different quantities of motion depending on what we know. For example, the equation for final velocity shows how speed changes with time and acceleration, while the equation for distance traveled relates initial velocity, acceleration, and time to determine how far the object moves. Other equations allow calculation of final velocity based on distance without knowing the time, or distance based on initial and final velocities. These tools are essential for analyzing any situation involving constant acceleration.

How is it that if acceleration is constant, the velocity increases by the same amount each second?

It makes sense when you think about it like this: constant acceleration means the object’s velocity grows by the same amount every second. For example, if an object gains 2 m/s every second, after 1 second it will have increased 2 m/s, after 2 seconds 4 m/s, and so on. This uniform increase makes the motion predictable and easy to calculate, even if it seems abstract at first.

Why does the distance formula include a term with time squared?

This happens because the object’s velocity is continuously changing due to acceleration. The squared time term represents the accumulated effect of this change over time, showing that distance does not increase linearly but grows faster as time passes. This is why the equations include both a linear term for initial velocity and a quadratic term for acceleration.

Does it make sense that there are multiple UARM equations to describe the same motion?

Yes, because each equation is designed to calculate specific variables depending on what is known and what we want to find. For instance, if you know initial velocity, acceleration, and time, one equation works perfectly. If you know distance and acceleration, another equation is more convenient. All equations describe the same type of motion, just from slightly different perspectives to solve problems more efficiently.