The Velocity Calculator finds final velocity, average velocity and the signed result of v² = u² + 2as under constant acceleration. Enter the known quantities for the mode that matches the problem, and the tool returns metres per second (or another selected speed unit) with the substituted arithmetic shown step by step.
Velocity is the quantity most problems ask for first. The same three rearrangements cover a car pulling away, a vehicle coming to a stop and the midpoint speed used to find distance when acceleration is steady.
Calculate final velocity from acceleration and time
Final velocity under constant acceleration follows from initial velocity, acceleration and elapsed time through one linear relation. Enter those three quantities and the tool returns the velocity at the end of the interval, converting units before the arithmetic so a problem stated in kilometres per hour still resolves correctly when the output is metres per second.
The defining relation is:
v = u + atRearrangements that solve for the other three variables:
u = v − at
a = (v − u)/t
t = (v − u)/aA cart starts at rest (u = 0) and accelerates at 2 m/s² for 3 s:
v = 0 + (2 × 3) = 6 m/s
Time must not be zero when solving for acceleration from this form. Acceleration must not be zero when solving for time if v differs from u. Both checks are stated in the result panel when they apply.
For the full set of four kinematic equations and the rule that picks among them, use the Kinematics Calculator. This page keeps the rearrangements that solve for velocity.
Calculate average velocity from initial and final
Average velocity under uniform acceleration is the arithmetic mean of the start and end velocities. That midpoint value multiplies by time to give displacement, which is why textbooks introduce it before the squared-time equation. The formula holds only while acceleration is constant; when acceleration varies, the true time-average of velocity is a different integral.
v̄ = (u + v) / 2With u = 5 m/s and v = 11 m/s:
v̄ = (5 + 11) / 2 = 8 m/s
Displacement over t = 3 s then follows as:
s = v̄ × t = 8 × 3 = 24 m
That product is identical to s = ½(u + v)t. Instantaneous velocity at a single clock reading is not this average. The average is a summary of the whole interval; the velocity-time graph is a straight line from u to v, and the average is the height of the rectangle with the same area as that trapezium.
Calculate velocity from acceleration and displacement
When time is unknown, final velocity still follows from initial velocity, acceleration and displacement. Square both sides of the linear velocity equation, eliminate t, and the relation that remains is v² = u² + 2as. Take the square root for v, and discard an imaginary result when the object never reaches that displacement.
v² = u² + 2as
v = ±√(u² + 2as)A vehicle braking from 10 m/s at −2 m/s² over 25 m:
v² = 10² + 2(−2)(25) = 100 − 100 = 0
v = 0 m/s
The object stops exactly at that displacement. If the quantity under the root is negative, no real final velocity exists for those inputs. Sign choice after the root follows the direction convention you set for the problem: the positive root for motion in the positive direction, the negative root when the object is moving the other way.
Find final velocity for u = 5, a = 2, t = 3
A car pulls away at 5 m/s and accelerates at 2 m/s² for 3 seconds. Final velocity is 11 m/s. The same numbers are the engine fixture for v = u + at, so the arithmetic below is what the tool must reproduce.
1. List what is known. u = 5 m/s, a = 2 m/s², t = 3 s. Displacement is not required for this unknown.
2. Apply the equation. v = u + at
3. Substitute. v = 5 + (2 × 3)
4. Solve. v = 5 + 6 = 11 m/s
Average velocity for the same interval:
v̄ = (5 + 11) / 2 = 8 m/s
With time known, displacement is 8 × 3 = 24 m. Checking with the no-time form:
v² = 5² + 2(2)(24) = 25 + 96 = 121, so v = 11 m/s again.
Distinguish speed from velocity
Speed is a scalar. Velocity is a vector. Both use the same SI unit of metres per second, which is why the words get swapped in casual speech, but the distinction matters as soon as direction changes. An object can keep a steady speed while its velocity changes continuously.
Circular motion at constant speed is the clean example. A car on a roundabout may hold 15 m/s on the speedometer while its velocity vector rotates with the turn. Centripetal acceleration exists even though the speed reading is flat. Straight-line problems hide the difference because direction is only forward or back, and the sign of v carries that information.
| Quantity | Type | What it records |
|---|---|---|
| Speed | Scalar | How fast, magnitude only |
| Velocity | Vector | How fast and which way |
Path length over time is average speed. Net change in position over time is average velocity. A walk five metres east and five metres west has average speed above zero and average velocity of zero. The Displacement Calculator owns that comparison in full; here the point is only that speed and velocity are not interchangeable labels.
Read the sign of a velocity
Positive velocity means motion in the direction you called positive. Negative velocity means motion the other way. The number alone never tells you which end of the road is positive; that choice is part of the problem setup and must stay fixed for every quantity in the same solution.
A ball thrown upward with up as positive leaves the hand at, say, +20 m/s, reaches zero at the peak, then returns with negative velocity. The acceleration due to gravity stays negative throughout if up is positive. Speed is the absolute value, so the ball is as fast on the way down as on the way up at the same height, while velocity has flipped sign.
Reversing during an interval is allowed under constant acceleration. The equations do not break when v crosses zero. What breaks the setup is mixing conventions mid-problem: calling right positive for u and left positive for a produces a sign error that looks like a wrong formula.
Convert between velocity units
Speed units differ by a fixed factor from metres per second. Convert every input to SI, solve, then convert the output if another unit is selected. The tool applies those factors automatically; the table below shows what the conversion layer uses.
| Unit | Factor to m/s |
|---|---|
| m/s | 1 |
| km/h | 1/3.6 ≈ 0.277778 |
| km/s | 1000 |
| mph | 0.44704 |
| ft/s | 0.3048 |
Worked check: 36 km/h in SI.
36 × (1000/3600) = 36 / 3.6 = 10 m/s
The same 10 m/s in miles per hour:
10 / 0.44704 ≈ 22.37 mph
In feet per second:
10 / 0.3048 ≈ 32.81 ft/s
Scientific notation such as 3.45e9 is accepted on every numeric field. Mixing units inside one problem without converting first is the usual source of wrong answers when working by hand; the unit selectors remove that step when every field is set before calculating.
Frequently asked questions
Final velocity, average velocity and the no-time root form raise the same cluster of exam questions: which formula fits the knowns, how sign works, and how speed differs from velocity. The answers below restate each question, then give the relation or distinction that settles it, with the fixture arithmetic where a number helps.
What is the formula for final velocity with acceleration and time?
Final velocity is initial velocity plus acceleration times time: v = u + at. The relation assumes constant acceleration. With u = 5 m/s, a = 2 m/s² and t = 3 s, v = 5 + 6 = 11 m/s.
How do you calculate average velocity?
Under constant acceleration, average velocity is (u + v) / 2. Multiply by time to get displacement. The formula is not the same as dividing total path length by time when the path doubles back; that quotient is average speed.
How do you find velocity without time?
Use v² = u² + 2as, then take the square root. Time cancels when the linear equation is squared and substituted. A negative value under the root means those inputs never occur together for real motion.
What is the difference between speed and velocity?
Speed is magnitude only. Velocity includes direction. Constant speed on a curve still means changing velocity because the direction of motion is changing. In one dimension the sign of v carries the direction.
Can velocity be negative?
Yes. A negative value means motion opposite the chosen positive axis. It does not mean the object is slowing down. Slowing down while moving in the positive direction is negative acceleration, which the Acceleration Calculator treats as the signed quantity it is.
Does average velocity equal half the final velocity?
Only when the object starts from rest and acceleration is constant. Then u = 0, so v̄ = v/2. If u is not zero, use (u + v) / 2 rather than v/2.
How do you convert km/h to m/s?
Divide by 3.6, because one kilometre per hour is 1000 metres in 3600 seconds. So 36 km/h = 10 m/s. Multiply by 3.6 to go the other way.
What if acceleration is zero?
Then v = u for the whole interval. Average velocity equals that constant value. Displacement is simply v × t. The squared form reduces to v² = u² as well.
Why does the square-root form give two signs?
Because squaring loses sign. Physically you restore the sign from the problem: motion in the positive direction takes the positive root, motion in the negative direction takes the negative root. Reporting both without a convention leaves the answer incomplete.
Is instantaneous velocity the slope of the displacement-time graph?
Yes. Under constant acceleration that graph is a parabola, and its gradient at any point is the velocity at that instant. The velocity-time graph is a straight line whose value at each moment is that same instantaneous velocity.
Summary
Final velocity follows from v = u + at, average velocity from (u + v) / 2 under uniform acceleration, and the no-time case from the root of v² = u² + 2as.
Worked arithmetic for u = 5 m/s, a = 2 m/s² and t = 3 s yields v = 11 m/s and an average of 8 m/s.
Speed and velocity share a unit but not a meaning: direction makes velocity a vector, and the sign of the result follows the axis you declare. Unit factors convert among m/s, km/h, mph and ft/s before and after the solve.