Velocity & Acceleration as Derivatives
Velocity and acceleration as derivatives extend motion analysis: acceleration is the derivative of velocity, \(a=\dfrac{dv}{dt}=\dfrac{d^{2}x}{dt^{2}}\).
Part of the NSW Year 12 Mathematics Advanced course, in the Calculus area of study (Applications of calculus focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
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.