Uniformly Accelerated Rectilinear Motion (UARM). Equations

09/04/2026

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.

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.

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

Equations
Fall
Free

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

Uniformly Accelerated Motion

This animation is an example of uniformly accelerated motion. The bus, starting from zero, is increasing its speed with constant acceleration.

Uniformly decelerated motion

This animation is an example of uniformly decelerated motion. The bus, starting from a high velocity, slows down with constant deceleration until it stops.

Uniformly Accelerated Motion

This animation shows the graphs of the motion of a car accelerating with constant acceleration from rest and maintaining a straight motion. What does the area under the line of the v – t graph represent?

Uniformly Decelerated Motion

This animation shows the graphs of the motion of a car decelerating with constant deceleration from its initial velocity to rest and then backing up.

Equations of motion

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

William Rowan Hamilton

1805

–

1865

William Rowan Hamilton developed Hamiltonian mechanics and quaternions, unifying geometry and mathematical physics

“Mathematics and physics meet in the harmony of the equations governing motion.”

Daniel Bernoulli

1700

–

1782

Daniel Bernoulli formulated Bernoulli’s principle and developed fundamental theories in fluid mechanics and gas dynamics

“Nature is always economical in its means”

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Joseph-Louis Lagrange

1736

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1813

Joseph-Louis Lagrange formulated the Lagrangian equations, key in classical mechanics to describe motion. He also founded variational calculus, applying it to algebra and mathematical physics

“As soon as a mathematical truth is established, it always serves as a starting point for new truth”

Gaspard-Gustave de Coriolis

1792

–

1843

Gaspard-Gustave de Coriolis formulated the effect that bears his name, explaining the deflection of bodies on Earth’s rotation and providing key foundations for mechanics.

“Motion is not only trajectory, it is also invisible influence”

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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.