Uniform Acceleration Calculator (Velocity & Displacement)
Calculate the final velocity and displacement of a uniformly accelerated linear motion from the initial velocity, acceleration, and elapsed time.
What motion at constant acceleration is
Motion at constant acceleration is **movement along a straight line whose acceleration never changes**. Because the acceleration holds steady, fixing three quantities — the initial velocity v0, the acceleration a and the elapsed time t — settles both the velocity and the position at that moment with a single formula each. This tool computes the final velocity as **v = v0 + at** and the displacement as **x = v0t + ½at²**. The units are fixed SI: metres per second, metres per second squared and seconds.
**Displacement x is not the distance travelled but the change in position, direction included.** In motion that turns back on itself — a ball thrown upward, say — the outward and return legs cancel, so the size of the displacement comes out smaller than the path actually covered. Initial velocity and acceleration each accept positive, negative and zero values, but **the elapsed time must be zero or greater.** To ask about an earlier moment, reset t = 0 to the instant the initial velocity was measured.
How to use the calculator
- Enter the initial velocity Give the starting speed in m/s. It is zero if the motion begins from rest.
- Enter the acceleration Give it in m/s². **Use a negative value to decelerate** — a = -3 for braking, for instance.
- Enter the elapsed time Give, in seconds, how far ahead you want the state. Negative values are refused.
- Read the velocity and displacement You get the final velocity and the displacement at that moment. A negative displacement means the object lies on the far side of where it started, against the direction of the initial velocity.
- Apply it to free fall Put the acceleration due to gravity — about 9.8 m/s² — into the acceleration field and you get the speed and the distance fallen from the moment of release.
Tips for getting more out of it
- To calculate a decelerating motion, enter a negative value for acceleration (a) — for example, a = -3 when braking a vehicle to slow it down.
- Displacement (x) represents the change in position including direction, not the total distance traveled. If acceleration and initial velocity point in opposite directions, displacement can even turn negative during back-and-forth motion.
- This tool also works for free fall. Enter Earth's gravitational acceleration (about 9.8 m/s²) as the acceleration to find the velocity and distance fallen at any elapsed time after release.
- Elapsed time (t) only accepts values of 0 or greater. If you need an instant before the reference point, redefine t=0 at the moment of the initial velocity and recalculate from there.
Where the calculator helps
Physics homework and exam practice
It serves for drilling the two formulas, and it suits checking an answer worked out by hand.
Estimating a braking distance
Put the travelling speed into the initial velocity and the negative deceleration into the acceleration, and you get a sense of the distance covered before stopping.
Checking the time and speed of a free fall
Set the acceleration to 9.8 m/s² and you can see how far something falls in a given number of seconds and how fast it is going.
Getting a feel for what the formulas say
Doubling the time roughly quadruples the displacement, because the t² term dominates. Changing the numbers makes such relations tangible.
When you also want angled motion and forces
For motion launched at an angle, see the projectile motion calculator; for work and energy, the work and kinetic energy calculator; for the effect of friction, the friction force calculator.
Terms about motion at constant acceleration
- Motion at constant acceleration
- Movement along a straight line at an acceleration that never changes. Free fall, and a car braking under steady force, are the standard examples.
- Initial velocity (v0)
- The velocity at the instant observation begins, t = 0. It is entered here in m/s and may be positive, negative or zero.
- Acceleration (a)
- The change in velocity per unit of time. A negative value means a force acting against the initial velocity, so the object slows.
- Displacement (x)
- The change in position, direction included. It differs from the path length: in motion that doubles back, it comes out smaller than the distance travelled.
- Acceleration due to gravity
- The acceleration of about 9.8 m/s² that a body near the earth’s surface undergoes. Free fall is treated as constant-acceleration motion with this figure as the acceleration.
- Distance travelled
- The total length actually covered. Since it adds up regardless of direction, it is always at least as large as the size of the displacement.
Frequently Asked Questions
Side Note — Galileo's Discovery of the Law of Falling Bodies
The concept of uniform acceleration is generally credited to the 17th-century Italian scientist Galileo Galilei. Before him, Aristotle's belief that heavier objects fall faster had been widely accepted for centuries. Through repeated experiments rolling balls down inclined planes, Galileo showed that a falling object's speed increases at a constant rate proportional to elapsed time, regardless of its mass.
Since accurate clocks did not yet exist, Galileo is said to have measured time using his own pulse and water clocks. Because rapid falls were too fast to measure precisely, he cleverly slowed the motion down by using gently inclined planes — an approach now celebrated as a pioneering example of experimental science.
This idea of uniformly accelerated motion later evolved into Isaac Newton's equation of motion (F = ma), forming the foundation of classical mechanics. Today, the formulas for uniform acceleration are still used across many fields of engineering, from calculating a car's braking distance to designing a rocket's launch trajectory.