Magnitude of Acceleration Calculator
Find the size (magnitude) of an acceleration vector from its x, y, z components — plus its direction angles — or from a change in velocity over time. Instant results in m/s², g-force, ft/s², and km/h/s.
|a| = √(ax² + ay² + az²)– m/s²
–g-force
–ft/s²
–km/h per second
– m/s²
–g-force
–ft/s²
–km/h per second
Worked Examples
| Input | Calculation | |a| |
|---|---|---|
| (3, 4, 0) m/s² | √(9 + 16 + 0) | 5.0 m/s² |
| (1, 2, 2) m/s² | √(1 + 4 + 4) | 3.0 m/s² |
| 0 → 100 km/h in 5 s | (27.78 − 0) / 5 | 5.56 m/s² ≈ 0.57 g |
Related: Angular Acceleration Calculator — for rotational (spin) acceleration instead of linear magnitude.
Frequently Asked Questions
What is the magnitude of acceleration formula?
The magnitude of an acceleration vector is
|a| = √(ax² + ay² + az²). For 2D motion the z-component is zero, so it simplifies to |a| = √(ax² + ay²). This is the Pythagorean theorem applied to the component arrows of the acceleration vector.How do I calculate acceleration from a change in velocity?
Use
a = (v₂ − v₁) / Δt: subtract the initial velocity from the final velocity and divide by the elapsed time in seconds. The magnitude is the absolute value of the result. Example: 0 to 100 km/h (27.78 m/s) in 5 seconds gives (27.78 − 0) / 5 = 5.56 m/s².How do I find the direction of the acceleration vector?
In 2D, the direction angle measured counter-clockwise from the +x axis is
θ = atan2(ay, ax). In 3D, direction is given by three angles (α, β, γ) from the +x, +y, +z axes, computed from the direction cosines: cos α = ax/|a|, cos β = ay/|a|, cos γ = az/|a|.What units does this calculator support?
Enter components in m/s². Results are shown in four units: m/s² (SI), g-force (divide by the standard gravity 9.80665 m/s²), ft/s² (multiply by 3.28084), and km/h per second (multiply by 3.6). For the Δv/Δt mode you can also enter velocities in km/h or mph.
Can the magnitude of acceleration be negative?
No. Magnitude is a length — it is always zero or positive. The individual components (ax, ay, az) can be negative, and the 1D acceleration a = (v₂ − v₁)/Δt can be negative (deceleration), but the magnitude |a| is the absolute value.
What is the direction when all components are zero?
If ax = ay = az = 0, the magnitude is 0 m/s² and the direction is undefined — there is no vector to point anywhere. This calculator reports the magnitude as 0 and leaves the direction angles blank.
What is the difference between magnitude of acceleration and angular acceleration?
Magnitude of acceleration measures how fast linear velocity changes (in m/s²). Angular acceleration measures how fast rotation speeds up or slows down (in rad/s²). For spinning objects, see our Angular Acceleration Calculator.
Is 1 g equal to 9.8 m/s²?
Approximately. The standard gravity used for g-force conversions is exactly 9.80665 m/s². So 5 m/s² equals 5 / 9.80665 ≈ 0.51 g. Engineers and physicists use 9.80665 as the standard, not 9.8, for precise conversions.
What is a real-world example of computing acceleration magnitude?
A car accelerating with ax = 3 m/s² forward and ay = 4 m/s² sideways (e.g. on a curve) has total acceleration magnitude √(3² + 4²) = 5 m/s², about 0.51 g. Its direction is atan2(4, 3) ≈ 53.1° from the forward axis.
Free tool by NRTechStudio