AcademyKinematics
Academy
Velocity
Level 1 - Math II (Physics) topic page in Kinematics.
Principle
Velocity measures how fast position changes and in which direction that change occurs. Average velocity compares two positions over a finite time interval. Instantaneous velocity is the derivative of position with respect to time.
Speed is the magnitude of velocity. Speed is a scalar, while velocity is a vector.
Notation
\(\mathbf r(t)\)
position vector
\(\mathrm{m}\)
\(\Delta t\)
elapsed time
\(\mathrm{s}\)
\(\mathbf v_{\text{avg}}\)
average velocity over a time interval
\(\mathrm{m\,s^{-1}}\)
\(\mathbf v(t)\)
instantaneous velocity
\(\mathrm{m\,s^{-1}}\)
\(|\mathbf v|\)
speed, the magnitude of velocity
\(\mathrm{m\,s^{-1}}\)
\(\dot x, \dot y, \dot z\)
time derivatives of the Cartesian position components
\(\mathrm{m\,s^{-1}}\)
Method
Step 1: Compute average velocity over an interval
Subtract positions and divide by elapsed time:
Average velocity
\[\mathbf v_{\text{avg}}=\frac{\Delta\mathbf r}{\Delta t}\]
Step 2: Take the instantaneous limit
When the time interval shrinks to zero, the average velocity becomes the derivative:
Instantaneous velocity
\[\mathbf v=\frac{d\mathbf r}{dt}\]
Step 3: Differentiate components
If \(\mathbf r(t)=x(t)\mathbf i+y(t)\mathbf j+z(t)\mathbf k\), differentiate each component with respect to time.
Start with position
\[\mathbf r(t)=x(t)\mathbf i+y(t)\mathbf j+z(t)\mathbf k\]
Differentiate with fixed Cartesian basis
\[\frac{d\mathbf r}{dt}=\frac{dx}{dt}\mathbf i+\frac{dy}{dt}\mathbf j+\frac{dz}{dt}\mathbf k\]
Use dot notation
\[\mathbf v=\dot x\mathbf i+\dot y\mathbf j+\dot z\mathbf k\]
Rules
Component velocity
\[\mathbf v=\dot x\mathbf i+\dot y\mathbf j+\dot z\mathbf k\]
Speed
\[\text{speed}=|\mathbf v|\]
Cartesian speed
\[|\mathbf v|=\sqrt{\dot x^2+\dot y^2+\dot z^2}\]
- Velocity has direction; speed does not.
- Average velocity depends only on net displacement and elapsed time.
- Instantaneous velocity is tangent to a smooth trajectory.
- The sign of a one-dimensional velocity records direction along the chosen axis.
Examples
Question
A particle moves from
\[\mathbf r_1=2\mathbf i\]
m to \[\mathbf r_2=8\mathbf i\]
m in \[3\]
s. Find the average velocity.Answer
The displacement is
\[\Delta\mathbf r=6\mathbf i\]
m. Divide by \[\Delta t=3\]
s to get \[\mathbf v_{\text{avg}}=2\mathbf i\,m s^{-1}\]
Checks
- Do not use distance travelled when calculating average velocity; use displacement.
- Do not call a negative one-dimensional velocity a negative speed. Speed is non-negative.
- Include direction or components when reporting velocity.
- Check units after differentiating: metres divided by seconds gives \(m s^{-1}\).