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\begin{large}\[ \vec{v} = \frac{d\vec{r}}{dt}\]\end{large}

\begin{large}\[ f(t)\]\end{large}

\begin{large}\[ \langle f(t)\rangle_{t} \equiv \frac{\int_{t_{i}}^{t_{f}} f(t)\:dt}{t_{f} - t_{i}} \]\end{large}

\begin{large}\[ \langle \vec{v} \rangle_{t} = \frac{\int_{t_{i}}^{t_{f}} \vec{v} \:dt}{t_{f}-t_{i}}
=\frac{\int_{t_{i}}^{t_{f}} \left(\frac{dx}{dt}\hat{x} + \frac{dy}{dt}\hat{y} + \frac{dz}{dt}\hat{z}\right) \:dt}{t_{f}-t_{i}}
\]\end{large}

\begin{large}\[ dxh2. Defnition 
{excerpt} 
The time rate of change of position.  Velocity is a vector quantity.  For one-dimensional motion, the direction is often specified by the mathematical sign of the velocity.  A positive velocity indicates motion in one (arbitrarily chosen) direction, while a negative velocity indicates the opposite direction.
{excerpt}

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h2. Representations

h4. Differential 
  {latex}\begin{Large} $v = \frac{dx}{dt}$\:dt\]\end{Large}{latex}
h4. Integral
 {latex}\begin{Large} $v_{\rm f} = v_{\rm i}+\int_{t_{\rm i}}^{t_{\rm f}}a\:dt$\end{Large}{latex}\\
{latex}\begin{Large}$v_{\rm f}^{2}=v_{\rm i}^{2}+2\int_{large}

\begin{large}\[ \langle \vec{v} \rangle_{t} = \frac{\int_{x_{i}}^{x_{f}} \hat{x}\:dx + \int_{y_{i}}^{y_{f}} \hat{y}\:dy
+ \int_{z_{i}}^{z_{f}} \hat{z}\:dz}{t_{f} - t_{i}} \]\end{large}

\begin{large}\[ \langle \vec{v} \rangle_{t} = \frac{\vec{r}_{f} - \vec{r}_{i}}{t_{f}-t_{i}} \equiv \frac{\Delta\vec{r}}{\Delta t} \]\end{large}

\begin{large} \[ \langle v\rangle_{t} = \frac{x_{\rm f} - x_{\rm i}}^{xt_{\rm f} - t_{\rm i}}a \:dx$] \end{Large}{latex}
h4. Graphical
  Besides explicit velocity graphs, velocity can be found from the slope of a position vs. time graph or (if the initial velocity is known) by adding the area under an acceleration vs. time graph to the initial velocity.
h4. Through Motion Diagrams
  In a motion diagram, the velocity can be estimated by looking at the spacing of the individual snapshots (assuming that the snapshots are separated by equal time intervals).

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h2. Relevant Models

 {children:page=Two-Dimensional Motion (General)|depth=all}

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h2. Relevant Examples

 {contentbylabel:1d_motion}

large}

\begin{large} \[ \langle v_{\rm pd}\rangle_{t} = \frac{x_{\rm p} - x_{\rm d}}{t_{\rm p} - t_{\rm d}} = \frac{ +2\:{\rm blocks} - 0\:{\rm blocks}}{2\:{\rm minutes}} = + 1\:{\rm blocks/min}\]\end{large}

One item that is worth noting is that physics problems often do not give actual times. Instead, they give elapsed times. In this situation, for instance, we were not told exactly when the student left the dorm (10:00 AM? 12:00 PM?) or when they arrived at the physics building. We were only told that the difference between the times was 2 minutes. (If the student left the dorm at 11:00 AM, they arrived at 11:02 AM). This information is sufficent to find the average velocity. Because it is so rare to be given initial and final times, the velocity equation is often written:

\begin{large}\[ \langle v \rangle_{t} = \frac{x_{\rm f} - x_{\rm i}}{t}\]\end{large}

where t denotes elapsed time.

It is a common source of confusion that the equations of mechanics often use "final" and "initial" as their subscripts. For the trip described at the beginning of this lesson, it is clear that the (overall) initial position is the dorm (x = 0 m) and the (overall) final position is the library (x = + 3 m), yet we have just used the equation for average velocity with the final position taken to be the physics building. The equations are not required to use the overall final and overall initial positions and times. You are free to break up the motion into as many segments as desired, and apply the equation to the beginning and end of each segment. The only requirement is that the position taken for the "initial" one occurs earlier in the motion than the "final" one.

\begin{large}\[ \langle v_{\rm cp}\rangle_{t} = \frac{x_{\rm c} - x_{\rm p}}{t_{\rm cp}} = \frac{(-1\:{\rm blocks})- (+ 2\:{\rm blocks})}{2\:{\rm minutes}} = -1.5\:{\rm blocks/min}\] \end{large}

Using both methods of reporting direction together in one statement results in confusion. What would it mean if we reported the student had an average velocity of - 1.5 blocks/min west?