Displacement, Velocity & Speed
Learn displacement, velocity and speed for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-06). A particle moving back and forth along a line has a position that changes over time, and calculus connects that position to how fast it is moving and in which direction.
You will learn to find velocity by differentiating displacement, interpret positive and negative values, find when the particle is at rest, and tell velocity from speed and displacement from total distance — the language of motion in physics.
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.