Velocity
Signed final velocity on the reference axis.
Physics · Mechanics · Motion & Energy
Constant acceleration motion calculator
Find final velocity and displacement from elapsed time, or use the time-free relation to find a positive final velocity or acceleration.
Created v1.0
By Thibaut Grzelak, Mechanical Analysis Engineer
02
Results
Signed displacement
Signed position change relative to the starting point.
Acceleration
Signed change in velocity per unit time.
Relation and numerical substitution
v = v₀ + a × t—
Constant acceleration; elapsed time ≥ 0; signed velocity and displacement.
01Formulas and symbols
Formulas used
v
v = v₀ + a × tSigned final velocity on the reference axis.Δx
Δx = v₀ × t + ½ × a × t²Signed position change relative to the starting point.v
v = √(v₀² + 2 × a × Δx)Signed final velocity on the reference axis.a
a = (v² − v₀²) / (2 × Δx)Signed change in velocity per unit time.02Assumptions and limits
Scope of validity
- Constant acceleration; elapsed time ≥ 0; signed velocity and displacement.
03Validation example
Reference numerical case
- v₀ = 5 m/s, a = 2 m/s², t = 4 s: v = 5 + 2 × 4 = 13 m/s; Δx = 5 × 4 + ½ × 2 × 4² = 36 m.
04References
Technical references
FAQ
Which kinematic equation should I use when time is unknown?
For constant acceleration, v² = v₀² + 2aΔx relates velocity, acceleration and displacement without using time. Use it only when the acceleration is constant over the interval.
Can I use these equations if acceleration changes with time?
Not over the whole motion as a single step. Variable acceleration requires integration, a more appropriate motion model, or a piecewise approximation over intervals where acceleration can reasonably be treated as constant.
05How it works
Under constant acceleration, velocity varies linearly with time and displacement quadratically. The time mode returns both quantities, including a reversal of direction. The time-free velocity mode selects only the positive square-root branch; it does not determine elapsed time.
06Limits of the model
One-dimensional motion with constant acceleration during the whole interval. Displacement is signed final position minus initial position, not total distance travelled. A negative radicand has no real solution; zero displacement cannot determine acceleration in the inverse mode.
07Common mistakes
Keep v₀, v, a and Δx on the same signed axis. Do not replace signed displacement with path length after a reversal. The positive time-free branch cannot represent a negative final velocity.