<p align="right">Last Update: <font color="#4f81bd">July, 22, 2024</font></p>
## BIG IDEAS
- Velocity is the rate of change in [[Notes Vault/Physics Notes Vault/Kinematics/1D Motion/Position|position]] with respect to [[time]].
- Symbol $\vec{v}$
- SI units: meters per second ($m/s$)
- Velocity is a [[Vectors|vector quantity]].
>[!important]
The / is a forward slash read as “per” and means _in each_.
### Formula
$v \ = \ \frac{\Delta x}{t} = \frac{x-x_0}{t} \tag{1}$
### Instantaneous vs. Average Velocity
<span style="background:#d3f8b6">Average velocity</span> is the total displacement divided by the total time taken.
$Average \ velocity \ = \ \frac{Total \ Displacement}{Total \ Time \tag{1}} $
<span style="background:#d3f8b6">Instantaneous Velocity</span> is the velocity of an object at a specific point in time; it is the derivative of the position with respect to time.
$Instantaneous \ velocity \ = \ \frac{dx}{dt} \tag{2}$
### Relationship to Displacement
The integral of velocity with respect to time gives the displacement. Displacement is a vector quantity that represents the change in position of an object over a period of time. Mathematically, this can be expressed as:
$\mathbf{s}(t) = \int \mathbf{v}(t) \, dt + \mathbf{s}_0 \tag{3}$
where:
- $\mathbf{s}(t)$ is the displacement as a function of time.
- $\mathbf{v}(t)$ is the velocity as a function of time.
- $\mathbf{s}_0$ is the initial displacement (position) at the starting time.
In simpler terms, the displacement is the area under the velocity-time graph, which accounts for the initial position as well. If the velocity is constant, the displacement can be calculated as:
$\mathbf{s} = \mathbf{v} \cdot t + \mathbf{s}_0 \tag{4}$
where $\mathbf{v}$ is the constant velocity, $t$ is the time interval, and $\mathbf{s}_0$ is the initial position. For non-constant velocity, the integral approach is necessary to find the exact displacement over the time interval.
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