Derivative of Velocity Calculator

Calculator

Calculation mode

Choose the input style that matches your data.

Velocity unit

Time unit

Decimal precision

Controls rounding in results and exports.

Two-point average acceleration

Uses: a = (v2 − v1) / (t2 − t1)

Formula used How to use

Formula used

Core definition

a(t) = dv(t) / dt

The derivative of velocity with respect to time is acceleration.

Average over an interval

a_avg = (v2 − v1) / (t2 − t1)

Use this when you only know two measurements.

Numerical differentiation (finite differences)

Forward: a ≈ [v(t+h) − v(t)] / h

Backward: a ≈ [v(t) − v(t−h)] / h

Central: a ≈ [v(t+h) − v(t−h)] / (2h)

Central difference is usually the most accurate for smooth data.

How to use this calculator

Select a calculation mode: two points, function, or dataset.
Pick the velocity and time units used by your inputs.
Enter your values, or paste the time–velocity pairs.
For functions, choose a small step size and scheme.
Click Calculate to see results above the form.
Use the export buttons to save results as CSV or PDF.

Example data table

Sample velocity measurements (m/s) at equal time steps (s). The middle rows use central differences.

Time (s)	Velocity v (m/s)	Approx. acceleration a (m/s²)	Difference scheme
0	0	3	Forward
1	3	4	Central
2	8	6	Central
3	15	7	Backward

If velocity follows v(t)=t²+2t, then the exact acceleration is a(t)=2t+2.

Derivative of velocity and acceleration overview

1) What the derivative means

Velocity describes how position changes with time. Its time-derivative, a(t)=dv/dt, tells how quickly velocity itself changes. Positive acceleration increases speed (or shifts direction), while negative acceleration reduces speed. In SI units, acceleration is measured in m/s².

2) Average versus instantaneous acceleration

With two measurements, the calculator uses a=(v2−v1)/(t2−t1). Example: a car goes from 0 to 100 km/h (27.78 m/s) in 6 s, so average acceleration is about 4.63 m/s². Instantaneous acceleration can vary during the run.

3) Function mode for smooth motion

If you know a velocity model like v(t)=t²+2t, the exact derivative is a(t)=2t+2. The calculator approximates this derivative using finite differences, which is useful when you do not want symbolic calculus, or when v(t) includes trig or exponential terms.

4) Why central difference is recommended

Central difference uses both sides of the point: [v(t+h)−v(t−h)]/(2h). For smooth functions, this typically reduces error compared with forward or backward schemes. As a quick rule, decreasing h improves accuracy until rounding limits appear.

5) Dataset mode for real measurements

Real sensors produce time–velocity pairs. This tool estimates acceleration at each timestamp using forward/backward differences at the ends and central differences in the middle (Auto mode). If times are uneven, acceleration still works, but repeated timestamps cause division by zero and must be fixed.

6) Units and quick conversions

The calculator converts your inputs internally and then returns results in your chosen units. Helpful facts: 1 km/h = 0.27778 m/s, 1 mph = 0.44704 m/s, and 1 min = 60 s. Consistent units prevent “hidden” scaling mistakes in acceleration.

7) Typical acceleration values

Gravity near Earth is about 9.81 m/s² downward. Comfortable passenger-car acceleration is often 1–3 m/s², while strong braking can reach 6–9 m/s². Sprint athletes may average around 3–4 m/s² early in a 100 m run, then acceleration tapers off.

8) Practical tips for stable results

For function mode, start with h near 0.001–0.01 (in your time unit) and compare schemes. For dataset mode, remove obvious outliers and keep a steady sampling interval when possible. If the velocity signal is noisy, consider averaging multiple readings before differentiating.

FAQs

1) Is the derivative of velocity always acceleration?

Yes. Acceleration is defined as the time derivative of velocity. If velocity is a vector, the derivative captures both speed changes and direction changes.

2) What is the difference between average and instantaneous acceleration?

Average acceleration uses two points over an interval. Instantaneous acceleration is the derivative at a specific time, often estimated with a small step size using finite differences.

3) Which difference scheme should I choose?

Use central difference for smooth functions because it is usually more accurate. For endpoints in datasets, forward or backward differences are necessary because one side is missing.

4) How small should the step size h be?

Choose h small enough to capture curvature but not so small that rounding dominates. Try 0.001–0.01 in your time unit, then compare results for stability.

5) Can I use km/h and minutes?

Yes. Select your velocity and time units. The tool converts internally and outputs acceleration in the combined unit (for example, km/h per min) for consistent reporting.

6) Why do I get “non-finite” or invalid results?

This happens if the function cannot be evaluated, the time step is zero, or your dataset contains repeated timestamps. Fix input values, ensure monotonic time, and try again.

7) Does this calculator replace a full physics model?

No. It computes acceleration from velocity information. Forces, mass, drag, and constraints still require separate modeling. Use the results as a measurement or a derived input.