Section 2.7 Velocity
Definition 2.7.1. Average Velocity.
The average velocity of an object is the object’s displacement divided by the interval of time required to move through that displacement:
\begin{equation*}
\vec{v}_{ave} = \frac{\Delta \vec{r}}{\Delta t}
\end{equation*}
When speaking to a friend or colleague we may not make a distinction between speed and velocity, but in physics the words are distinct. Where velocity is a vector, indicating both magnitude and direction, the word speed refers only to the magnitude of the velocity.
Physics also distinguishes between average quantities for intervals of definite time, and instantaneous quantities for infinitesimal intervals. To signal the difference, you will use the symbol \(d\) for an infinitesimal quantity instead of the \(\Delta\) symbol that you use for a definite change.
Definition 2.7.3. Instantaneous Velocity.
The instantaneous velocity of an object is given by
\begin{equation*}
\vec{v} = \frac{d \vec{r}}{d t}
\end{equation*}
Note that when you make this change in notation, the velocity suddenly becomes a derivative!
Exercises Activities
With definitions like the one above, you should aim to understand each mathematical symbol in the symbolic equation, along with how those symbols work together. You should also know the kind of quantity each symbol represents, and the units associated with that quantity. The exercises below are intended to help you build this understanding.
1. Summarize What You Learned - Velocity.
Write a 1-2 sentence description of what the definition of average velocity says in words.
2. Sensemaking: Units.
What are the units of velocity? Explain how these units are consistent with the units of the other quantities in the definition of average velocity given above.
For the next two exercises, make sure to remember all the steps for Explanations in Physics from
Figure 2.1.2.
3. Explanation: Direction of Velocity.
How does the direction of the average velocity vector compare to the direction of the displacement vector? Explain your reasoning.
Answer.The average velocity vector points in the same direction as the displacement vector and tells us the direction of the object’s motion.
4. Explanation: Velocity or Speed.
In physics we will use both the idea of velocity and the idea of speed. In everyday language, these two words are often used interchangeably, but in physics they have distinct meanings. Write 1-2 sentences aimed at explaining the difference between velocity and speed to a child.
