Velocity
The time rate of change of position.
Mathematical Definition
\begin
[ \vec
= \frac{d\vec{r}}
]\end
Representing Velocity Graphically
Velocity is commonly represented graphically in several ways:
- On a [velocity versus time graph].
- As the spacing between points in a motion diagram.
- As the slope of a position versus time graph.
Average Velocity
Mathematical Definition
In mechanics, the term "average velocity" will almost always be used to denote the time-averaged velocity. The general defnition of the time average of a function
\begin
[ f(t)]\end
is:
\begin
[ \langle f(t)\rangle_
\equiv \frac{\int_{t_{i}}^{t_{f}} f(t)\:dt}{t_
- t_{i}} ]\end
.
In the special case of velocity, this expression becomes:
\begin
[ \langle \vec
\rangle_
= \frac{\int_{t_{i}}^{t_{f}} \vec
\:dt}{t_
-t_{i}}
=\frac{\int_{t_{i}}^{t_{f}} \left(\frac
\hat
+ \frac
\hat
+ \frac
\hat
\right) \:dt}{t_
-t_{i}}
]\end
We can now formally split the numerator into three integrals and make a change of variables in each of the integrals. Noting that (by the chain rule):
\begin
[ dx = \frac
\:dt]\end
with the corresponding expressions for dy and dz, we have:
\begin
[ \langle \vec
\rangle_
= \frac{\int_{x_{i}}^{x_{f}} \hat
\:dx + \int_{y_{i}}^{y_{f}} \hat
\:dy
+ \int_{z_{i}}^{z_{f}} \hat
\:dz}{t_
- t_{i}} ]\end
These integrals are extremely simple, and lead to the very simple final expression:
\begin
[ \langle \vec
\rangle_
= \frac{\vec
_
- \vec
{i}}{t
-t_{i}} \equiv \frac{\Delta\vec{r}}
]\end
Thus, the average velocity is simply the total change in position for a trip divided by the total elapsed time for the trip.
A One-Dimensional Example
Consider an example of one-dimensional motion. Suppose a student rushes from their dorm to the physics building in 2 minutes. After spending 4 minutes turning in their homework, the student hurries to the cafeteria in 2 minutes. The student eats lunch for 12 minutes, then walks to the library in 6 minutes. During which portion of the trip was the student moving the fastest?
For simplicity, imagine a school where all these buildings are on the same street. The street runs east to west. Suppose that the physics building is two blocks east of the dorm, the cafeteria is one block west of the dorm, and the library is three blocks east of the dorm. Before performing any calculations to characterize this trip, it is necessary to set up a coordinate system. One possibility, which we will use in this example, is shown below.
null
We are now ready to return to our question: in which portion of the trip was the student moving the fastest? The average velocity for a trip in one dimension will be defined as:
\begin
[ \langle v\rangle_
= \frac{x_
- x_{\rm i}}{t_
- t_{\rm i}} ] \end
To see how this equation works, consider the first part of the student's trip. In that part, the student moved from the dorm to the physics building in a time of 2 minutes. To evaluate the average velocity for this part, we simply substitute into the equation:
\begin
[ \langle v_
\rangle_
= \frac{x_
- x_{\rm d}}{t_
- t_{\rm d}} = \frac{ +2\:
- 0\:{\rm blocks}}{2\:{\rm minutes}} = + 1\:
]\end
where we have used the subscript "p" to stand for the physics building and "d" for the dorm.
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
[ \langle v \rangle_
= \frac{x_
- x_
}
]\end
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.
We can compare this to the average velocity for the second trip made (from the physics building to the cafeteria):
\begin
[ \langle v_
\rangle_
= \frac{x_
- x_{\rm p}}{t_{\rm cp}} = \frac{(-1\:
)- (+ 2\:
)}{2\:{\rm minutes}} = -1.5\:
] \end
The first thing to note here is that our answer has come out with a negative sign. For the first leg of the trip, the student has a velocity of + 1 block/min, and for the second leg, a velocity of - 1.5 blocks/min. These signs indicate the direction of the students motion, just as the sign of the position difference did. When reporting average velocities, it is a good practice to explicitly give the meaning of the signs, so that people do not have to be familiar with your specific coordinate system to understand the result. Thus, in this case, a more general way to report the student's movement is to say that for the first leg the average velocity was 1 block/min east, and for the second leg the average velocity was 1.5 blocks/min west. When the direction is included, the sign is removed.
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?