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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 2.5 Acceleration

The term acceleration describes the rate of change of an object’s velocity. Like velocity (the rate of change of an object’s position), acceleration can be either average (involving large changes written using \(\Delta\)) or instantaneous (involving small, infinitesimal changes written using \(d\)).

Definition 2.5.1. Average Acceleration.

The average acceleration of an object is the object’s change in velocity divided by the interval of time required to change the velocity:
\begin{equation*} \vec{a}_{ave} = \frac{\Delta \vec{v}}{\Delta t} \end{equation*}

Definition 2.5.2. Instantaneous Acceleration.

The instantaneous acceleration of an object is found by taking the limit as \(\Delta t \rarrow 0\text{,}\) resulting in infinitesimal changes in velocity and time:
\begin{equation*} \vec{a} = \frac{d \vec{v}}{d t} \end{equation*}

Exercises Activities

1. Summarize What You Learned - Acceleration.

Write a 1-2 sentence description of what the definition of acceleration says in words.

2. Sensemaking: Units.

What are the units of acceleration? Explain how these units are consistent with the units of other quantities you have learned about.
Answer.
m/s\(^2\)

3. Explanation: Direction.

How does the direction of the average acceleration vector compare to the direction of the change in velocity vector?
Tip.
Does acceleration have to point in the same direction as velocity?
Answer.
The average acceleration vector points in the same direction as the change in velocity!
Acceleration is a vector in the same direction as the change in velocity \(\Delta v\text{.}\) Since velocity is a vector, it can change in magnitude or direction (or both). Acceleration, therefore, results when there is a change in either speed or the direction of motion (or both).

Note 2.5.3. Deceleration.

Keep in mind that although acceleration is in the direction of the change in velocity, it is not always in the direction of motion. When an object slows down, its acceleration is opposite to the direction of its motion. Although this is referred to as deceleration in everyday language, the term deceleration can cause confusion, so physics does not use it. Instead, acceleration is a vector, so it is best characterized with a direction, most often with an appropriate sign in your chosen coordinate system. For example, in the case of the train in the figure below, which is slowing down, you might choose the velocity to be in the positive direction and the acceleration to be in the negative direction. In this case, you would say the train is undergoing negative acceleration, not deceleration. Alternatively, you might decide the velocity is in the negative direction and the acceleration is in the positive direction, which would still correspond to the train slowing down.
Figure 2.5.4. A subway train in Sao Paulo, Brazil, accelerates opposite to the motion as it comes into a station. It is accelerating in a direction opposite to its direction of motion. (Credit: modification of work by Yusuke Kawasaki)

References References

[1]
Some text and the Train Acceleration Figure adapted from OpenStax: https://openstax.org/books/university-physics-volume-1/pages/3-3-average-and-instantaneous-acceleration.
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