Displacement, Velocity & Speed
Displacement, velocity and speed describe motion in a straight line, with velocity the derivative of displacement, \(v=\dfrac{dx}{dt}\), and speed its size.
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
Velocity is the derivative of displacement; speed is its size. This Year 12 Mathematics Advanced topic (MAV-12-06) analyses straight-line motion.
\(v=\dfrac{dx}{dt}\) is the velocity (with direction, so a sign). Speed \(=|v|\). The particle is at rest when \(v=0\).
Method
- Differentiate \(x\) to get \(v\).
- Speed is \(|v|\).
- At rest when \(v=0\).
| \(v\) | \(=\) | \(2t\) |
| \(v(3)\) | \(=\) | \(6\ \text{m/s}\) |
| \(\text{speed}\) | \(=\) | \(|-8|=8\ \text{m/s}\) |
| \(2t-6\) | \(=\) | \(0\Rightarrow t=3\ \text{s}\) |
| \(v\) | \(=\) | \(3t^{2}-3\) |
Common pitfalls
Frequently asked questions
What is the difference between velocity and speed?
Velocity has direction and a sign; speed is its size (absolute value), so a velocity of minus 8 is a speed of 8.
How do you find velocity from displacement?
Differentiate the displacement with respect to time.
When is a particle at rest?
When its velocity is zero, not when its displacement is zero.
What does a negative velocity mean?
The particle is moving in the negative direction.