🚀 Acceleration Calculator, Solve a = (vf - vi) / t

What is the Acceleration Calculator?

Acceleration describes how quickly an object changes its velocity. If a car goes from a standstill to 27 metres per second in 9 seconds, its velocity changed by 27 m/s over 9 seconds, which is an average acceleration of 3 m/s every second, written 3 m/s squared. The defining equation for average acceleration is a = (vf - vi) / t, where vf is the final velocity, vi is the initial velocity, and t is the elapsed time. The numerator (vf - vi) is the change in velocity, often written as delta v. This single relationship is the workhorse of introductory mechanics and shows up in homework, driving theory, sports science, and engineering alike.

Because this is a single equation linking four quantities, you can rearrange it to solve for whichever one is unknown. To find the final velocity use vf = vi + a times t. To find the initial velocity use vi = vf - a times t. To find the time use t = (vf - vi) / a. The calculator above does this rearranging for you: pick the variable you want from the "Solve for" menu and the matching box is filled in automatically once you supply the other three numbers. There is no algebra to remember, but knowing the rearrangements helps you check that the answer is sensible.

There is more than one way to reach acceleration, and competing methods are worth knowing. From Newton second law you can use a = F / m, where F is the net force in newtons and m is the mass in kilograms, which is the right tool when you know the pushing or pulling force rather than the times. When you know the distance travelled but not the time, the kinematic equation vf squared = vi squared + 2 times a times s rearranges to a = (vf squared - vi squared) / (2 times s), where s is the displacement. Pick the formula that matches the data you actually have: velocity and time, force and mass, or velocity and distance.

Sign matters in acceleration problems. A positive acceleration means the object is speeding up in the direction you have chosen as positive, while a negative value (sometimes called deceleration or retardation) means it is slowing down or speeding up in the opposite direction. For example, a car braking from 30 m/s to a stop has a negative acceleration because its velocity is decreasing. Always keep your initial and final velocities measured in the same direction so the subtraction (vf - vi) carries the correct sign. Acceleration is a vector, so direction is as much a part of the answer as the number.

Keep your units consistent. The formula returns m/s squared only when velocities are in metres per second and time is in seconds. If your speed is given in kilometres per hour, convert it first (divide by 3.6 to get metres per second), and if your time is in minutes or hours convert it to seconds. To express the result in feet per second squared, multiply m/s squared by 3.281, and to express it in g (multiples of standard gravity) divide by 9.81. Always convert before you calculate, never after, to avoid compounding rounding errors.

Note that this tool computes average acceleration over an interval, the total change in velocity divided by the total time. When acceleration is constant (as in free fall near the surface of the Earth, about 9.81 m/s squared) the average value equals the instantaneous value at every moment, so the two are identical. When the acceleration varies during the interval, the result is the average across the whole interval rather than the value at any single instant. For changing acceleration, instantaneous acceleration is the derivative of velocity with respect to time, but for the constant case this calculator covers, the simple difference quotient is exact.

When to use it

• Solving physics homework where you must find acceleration, a velocity, or the time from the formula a = (vf - vi) / t.
• Working out how fast a vehicle accelerates from rest to a known speed over a measured time, such as a 0 to 60 mph time.
• Calculating the deceleration of a braking car or train so you can compare it against safe stopping limits.
• Finding the final speed of an object after a known acceleration acts on it for a given number of seconds.
• Converting a car spec sheet 0 to 100 km/h time into an average acceleration in m/s squared or in g.
• Checking sports and fitness data, such as a sprinter or a rollercoaster, by turning velocity changes into g-force.

How to use the Acceleration Calculator

1. Choose what you want to solve for from the "Solve for" menu: acceleration, final velocity, initial velocity, or time.
2. Convert your inputs first if needed: speeds to metres per second (km/h divided by 3.6) and time to seconds.
3. Enter the three values you already know in metres per second and seconds.
4. Read the answer in the result panel, along with the change in velocity (vf - vi).
5. Use the Copy result button to grab the full set of values, or the sample buttons to load a worked scenario.

Formula & method

Worked examples

Rearrangements of a = (vf - vi) / t

Reference accelerations for comparison

Acceleration unit conversions

Common mistakes to avoid

• Mixing up initial and final velocity. The change in velocity is final minus initial, vf - vi, in that order. Swapping them flips the sign of the acceleration, turning speeding up into slowing down. Decide which value is the starting speed before you subtract.
• Leaving speeds in km/h. The formula gives m/s squared only when velocities are in metres per second. If a speed is in kilometres per hour, divide by 3.6 first. For example 90 km/h becomes 25 m/s before you put it into the calculation.
• Dividing by zero time. You cannot find acceleration over zero seconds, because a = (vf - vi) / t would divide by zero. If you are solving for time instead, make sure the acceleration is not zero, otherwise the velocity never changes and no finite time exists.
• Dropping the negative sign for deceleration. When an object slows down, vf is smaller than vi, so the acceleration comes out negative. That minus sign is meaningful: it tells you the object is decelerating. Reporting only the size loses the direction of the change.
• Confusing acceleration with velocity or speed. Velocity is how fast something moves; acceleration is how fast that velocity changes. An object can have a high speed and zero acceleration (cruise control) or zero speed and large acceleration (the instant a ball reverses at the top of a bounce).
• Using the wrong formula for the data you have. Use a = (vf - vi) / t when you know velocities and time, a = F / m when you know force and mass, and a = (vf^2 - vi^2) / (2s) when you know velocities and distance but not the time. Picking the formula that matches your knowns avoids needless conversions.

Glossary

Acceleration (a)

The rate of change of velocity with time, measured in metres per second squared (m/s squared). It is a vector, so it has both size and direction.

Initial velocity (vi)

The velocity of the object at the start of the time interval being measured, sometimes written u.

Final velocity (vf)

The velocity of the object at the end of the time interval being measured, sometimes written v.

Change in velocity (delta v)

The difference vf minus vi, the amount by which the velocity changed over the interval.

Deceleration

A negative acceleration, where the object slows down because its velocity decreases over time. Also called retardation.

Average acceleration

The total change in velocity divided by the total time, as opposed to the instantaneous value at a single moment.

Instantaneous acceleration

The acceleration at one exact moment, equal to the derivative of velocity with respect to time. It matches the average value when acceleration is constant.

g-force

Acceleration expressed in multiples of standard gravity, where 1 g equals 9.81 m/s squared. Used to describe forces felt by people in vehicles, aircraft, and rides.

Frequently asked questions