Velocity & Acceleration as Derivatives
Learn velocity and acceleration as derivatives for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-06). Differentiating displacement gives velocity, and differentiating again gives acceleration, so one formula for position describes a particle’s whole motion.
You will learn to find acceleration as the second derivative of displacement, evaluate it at given times, and interpret the signs of velocity and acceleration for particles and projectiles — essential kinematics.
Theory
Acceleration is the derivative of velocity and the second derivative of displacement. This Year 12 Mathematics Advanced topic (MAV-12-06) links the three.
\(v=\dfrac{dx}{dt}\) and \(a=\dfrac{dv}{dt}=\dfrac{d^{2}x}{dt^{2}}\) (also written \(\ddot{x}\)). Differentiate displacement twice, or velocity once, to get acceleration.
Method
- Differentiate \(x\) once for \(v\).
- Differentiate again for \(a\).
- Substitute the time if a value is asked.
| \(v\) | \(=\) | \(3t^{2}\) |
| \(a\) | \(=\) | \(6t\Rightarrow a(2)=12\) |
| \(v\) | \(=\) | \(2t+3\) |
| \(a\) | \(=\) | \(2\) |
| \(a\) | \(=\) | \(8t\Rightarrow a(1)=8\) |
| \(v\) | \(=\) | \(3t^{2}-6t+2\) |
| \(a\) | \(=\) | \(6t-6\) |
Common pitfalls
Frequently asked questions
What is acceleration in terms of derivatives?
It is the derivative of velocity, and the second derivative of displacement, written d squared x by dt squared or x double dot.
How do you find acceleration from displacement?
Differentiate the displacement twice.
What is the notation for acceleration?
It is written d squared x by dt squared, or with two dots over x.
If velocity is constant, what is the acceleration?
Zero, because the velocity is not changing.