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Average acceleration is the change in velocity divided by the time it took: \(a = \frac{v – v_0}{t}\). A car that speeds up from 0 to 27 m/s in 6 seconds has \(a = \frac{27 – 0}{6} = 4.5\ \text{m/s}^2\). If you know the position \(s(t)\) as a formula, acceleration is its second derivative: \(a(t) = s”(t)\).
Three Ways to Find Acceleration
Average Acceleration Example
A car goes from 0 to 27 m/s (about 60 mph) in 6 seconds.
\[a = \frac{27 – 0}{6} = 4.5\ \text{m/s}^2.\]
That means the car gains 4.5 m/s of speed every second. If the acceleration is steady, it also covers \(d = \tfrac12 a t^2 = \tfrac12(4.5)(6)^2 = 81\) meters in that time.
| Velocity unit | Time unit | Acceleration unit |
|---|---|---|
| m/s | s | m/s² |
| ft/s | s | ft/s² |
| km/h | s | km/h per second |
Mixing units is the number one mistake. Convert km/h to m/s by dividing by 3.6 before you use the formula.
Calculus Version: Position → Velocity → Acceleration
When position is a function of time, each derivative tells you something new:
Example: \(s(t) = t^3 – 6t^2 + 9t\) meters.
- Velocity: \(v(t) = 3t^2 – 12t + 9\)
- Acceleration: \(a(t) = 6t – 12\)
- At \(t = 4\): \(v(4) = 9\) m/s and \(a(4) = 12\) m/s²
The derivative calculator shows the power rule steps for each derivative. Going the other way, from acceleration back to velocity and position, means taking an antiderivative.
Free Fall: The Famous 9.8
An object dropped near Earth’s surface falls a distance of about \(s(t) = 4.9t^2\) meters (ignoring air). Then \(v(t) = 9.8t\) and \(a(t) = 9.8\ \text{m/s}^2\). The acceleration is the same at every moment. That constant is \(g\), the acceleration due to gravity. In feet, the same rule is \(s(t) = 16t^2\) and \(g \approx 32\ \text{ft/s}^2\).
Speeding Up or Slowing Down?
A negative acceleration does not always mean slowing down. Compare the signs of velocity and acceleration:
| Velocity | Acceleration | The object is… |
|---|---|---|
| + | + | Speeding up |
| + | − | Slowing down |
| − | − | Speeding up (going faster backward) |
| − | + | Slowing down |
The calculator checks this for you when you enter a time.
Learn the Math Behind the Acceleration Calculator
Acceleration is the derivative of velocity. Going the other way, integrating acceleration gives velocity, and integrating velocity gives position.
Frequently Asked Questions
What is the difference between speed, velocity and acceleration?
Speed is how fast something moves. Velocity is speed with a direction (it can be negative). Acceleration is how fast the velocity changes.
Can acceleration be negative?
Yes. Negative acceleration means the velocity is decreasing. Whether the object slows down depends on the sign of the velocity too, as the table above shows.
What is instantaneous acceleration?
It is the acceleration at one exact moment, found with a derivative: \(a(t) = v'(t)\). Average acceleration covers a whole time interval. The instantaneous rate of change calculator explains the difference with a limit table.
How do I find the time it takes to stop?
Set the final velocity to 0 and solve \(0 = v_0 + at\) for \(t\): \(t = -v_0/a\). A car going 20 m/s that brakes at \(-5\ \text{m/s}^2\) stops in 4 seconds.
Related Calculators
- Derivative calculator
- Second derivative calculator
- Integral calculator to go from acceleration back to position