Velocity is the rate of change of position (displacement) with time, including direction. The method for how to find velocity depends on whether an average value or an instantaneous value is needed, and on what information is given (data, equations, or graphs).
Step 1: Distinguish velocity from speed
Speed is a scalar (no direction). Velocity is a vector (direction matters), so signs and directions (east/west, up/down, positive/negative axis) must be included.
Typical SI units: \( \text{m/s} \). In one dimension, direction is often represented by a sign.
Step 2: Find average velocity from displacement and time
Average velocity over a time interval is defined by displacement divided by elapsed time:
\[ v_{\text{avg}}=\frac{\Delta x}{\Delta t}=\frac{x_2-x_1}{t_2-t_1}. \]Use displacement \(\Delta x\), not total distance traveled. If an object moves out and back, \(\Delta x\) may be small (or zero) even if the distance is large.
Example (average velocity): A cart moves from \(x_1=10\,\text{m}\) to \(x_2=160\,\text{m}\) in \(t_2-t_1=12\,\text{s}\). \[ v_{\text{avg}}=\frac{160-10}{12}=\frac{150}{12}=12.5\,\text{m/s}. \]
Step 3: Find instantaneous velocity (velocity at a specific moment)
Instantaneous velocity is the time derivative of position:
\[ v(t)=\frac{dx}{dt}. \]With measured data, instantaneous velocity at a time is approximated by using a very small time interval around that time:
\[ v(t)\approx \frac{\Delta x}{\Delta t}\quad \text{for a sufficiently small }\Delta t. \]Step 4: Use kinematic equations when acceleration is constant
If motion is one-dimensional with constant acceleration \(a\), velocity can be found without calculus using standard kinematics:
| Given | Useful velocity relationship | When it helps |
|---|---|---|
| \(v_0\), \(a\), \(t\) | \(v = v_0 + a t\) | Velocity after time \(t\) |
| \(v_0\), \(a\), \(\Delta x\) | \(v^2 = v_0^2 + 2 a \Delta x\) | Velocity after moving a known displacement |
| \(v\), \(v_0\), \(t\) | \(a=\dfrac{v-v_0}{t}\) | Back-solving when acceleration is unknown |
Example (constant acceleration): An object starts from rest (\(v_0=0\)) and accelerates at \(a=2.5\,\text{m/s}^2\) for \(t=8\,\text{s}\). \[ v = v_0 + a t = 0 + (2.5)(8)=20\,\text{m/s}. \]
Step 5: Find velocity from motion graphs
Graphs provide velocity through slopes and areas:
Additional graph facts: the slope of a velocity–time graph equals acceleration \(a\), and the area under a velocity–time graph equals displacement \(\Delta x\).
Step 6: Common checks to avoid mistakes
| Check | What to verify |
|---|---|
| Direction/sign | Negative velocity indicates motion in the negative axis direction. |
| Units | Convert time to seconds and distance to meters before finalizing \( \text{m/s} \). |
| Displacement vs distance | \(\Delta x\) uses final minus initial position; it is not total path length. |
| Model choice | Use kinematics only when acceleration is constant; otherwise use data/graphs or \(v=dx/dt\). |
Final summary
To find velocity, use \(v_{\text{avg}}=\Delta x/\Delta t\) for an interval, \(v=dx/dt\) for an instant, or kinematic equations such as \(v=v_0+at\) and \(v^2=v_0^2+2a\Delta x\) when acceleration is constant; from graphs, velocity is the slope of the position–time curve and displacement is the area under the velocity–time curve.