AcademyOne-Dimensional Motion
Academy
Instantaneous Velocity
Level 1 - Physics topic page in One-Dimensional Motion.
Principle
Instantaneous velocity is the limiting slope of position at one time. It keeps the sign of motion along the chosen axis.
Notation
\(v_x\)
instantaneous velocity
\(\mathrm{m\,s^{-1}}\)
\(x(t)\)
position function
\(\mathrm{m}\)
\(\Delta x\)
small displacement
\(\mathrm{m}\)
\(\Delta t\)
small time interval
\(\mathrm{s}\)
Method
Shrink the interval
\[v_x = \lim_{\Delta t\to 0}\frac{x(t+\Delta t)-x(t)}{\Delta t}\]
Differentiate
\[v_x = \frac{dx}{dt}\]
Speed
\[v = |v_x|\]
Speed removes direction; velocity keeps it.
The tangent line below is the local linear model of the position graph near one instant. Its slope is the instantaneous velocity at that instant.
If the tangent is horizontal, the velocity is zero at that instant; the particle may still reverse direction afterward.
Rules
These are the compact results from the method above.
Limit definition
\[v_x = \lim_{\Delta t\to 0}\frac{\Delta x}{\Delta t}\]
Derivative form
\[v_x = \frac{dx}{dt}\]
Speed
\[v = |v_x|\]
Examples
Question
For
\[x(t)=3t^2-2t\]
find the velocity at \[t=2\,\text{s}\]
Answer
\[v_x=\frac{dx}{dt}=6t-2\]
\[v_x(2)=10\,\text{m s}^{-1}\]
Checks
- Positive velocity follows the positive axis.
- Zero velocity can occur while moving turns around.
- Steeper position graphs mean larger speed.