Average acceleration measures how quickly velocity changes over a given time interval. It is the most common form used in introductory physics:
Step-by-step example: A cyclist speeds up from 5 m/s to 15 m/s in 4 seconds.
Average acceleration assumes uniform (constant) acceleration over the interval. Instantaneous acceleration at a single moment requires the derivative: a = dv/dt. When acceleration varies continuously, velocity can be recovered by integration. For the reverse operation (finding position from velocity or velocity from acceleration), the Antiderivative Calculator demonstrates how integration works step by step.
| Scenario | vi (m/s) | vf (m/s) | t (s) | a (m/s²) |
|---|---|---|---|---|
| Car from rest to 60 km/h | 0 | 16.67 | 6 | 2.78 |
| Sprinter over 100 m | 0 | 10.4 | 10 | 1.04 |
| Braking car | 25 | 0 | 5 | -5.0 |
| Elevator reaching cruising speed | 0 | 3 | 2 | 1.5 |
| Falling object (gravity) | 0 | 9.81 | 1 | 9.81 |
Newton's second law gives the relationship between the net force acting on an object, its mass, and the resulting acceleration. The formula can be rearranged for any of the three variables:
Key requirement: F must be the net force, which is the vector sum of all forces on the object. In real problems this means subtracting friction, air resistance, and other opposing forces from the applied force before dividing by mass.
| Object | Net Force (N) | Mass (kg) | Acceleration (m/s²) |
|---|---|---|---|
| Person pushing a box | 120 | 40 | 3.0 |
| Car engine braking | -8,000 | 1,500 | -5.3 |
| Rocket at launch | 30,000,000 | 2,000,000 | 15.0 |
| Ball thrown upward | -0.49 (gravity) | 0.05 | -9.81 |
| Motor driving a shaft | 200 | 8 | 25.0 |
In multi-dimensional problems, force is a vector and acceleration has both magnitude and direction. Solving for components of acceleration in two or three dimensions uses the same a = F/m formula applied separately to each axis. For matrix-based methods used in multi-body physics systems, the Matrix Determinant Calculator covers the linear algebra tools used in those calculations.
The five kinematic equations let you find velocity, distance, or time from any combination of known values, provided acceleration is constant:
| Equation | Use when you know | Solves for |
|---|---|---|
| vf = vi + at | vi, a, t | Final velocity |
| d = (vi + vf) / 2 x t | vi, vf, t | Distance |
| d = vi x t + (1/2) x a x t² | vi, a, t | Distance |
| vf² = vi² + 2 x a x d | vi, a, d | Final velocity (no time needed) |
| d = vf x t - (1/2) x a x t² | vf, a, t | Distance |
Worked example (no time given): A ball rolls from rest down a ramp with acceleration 2 m/s² and travels 18 m. Find the final velocity.
These equations hold only for constant acceleration. When acceleration varies with time, the correct approach is calculus: velocity is the integral of acceleration over time, and position is the integral of velocity. For experimental physics data where you want to check whether acceleration correlates with another variable, the Correlation Coefficient Calculator computes Pearson r for two data sets.
A car accelerates from rest (0 m/s) to 27.8 m/s (100 km/h) in 8 seconds. Find the acceleration and distance covered.