Chapter 4: Measuring motion

4.4 Acceleration

If you closed your eyes and were transported to another location, you would not know what your new position is; we do not have a sense of position. If you closed your eyes and were moved at a constant speed, you also would not know. We do not have a sense of velocity, either. However, if your speed changed, or if your direction of motion changed, you would know—we have a sense that detects changes in motion. This sense is how we stay balanced and is detected by organs in the inner ear. Large or sudden changes in motion may also be felt in your gut.

Just like velocity is the rate of change of displacement, acceleration is the rate of change of velocity:

\[
\text{acceleration} = \frac{\text{change in velocity}}{\text{change in time}}
\]

Or, in symbols (for motion along the \(x\) axis):

\[
a_x = \frac{\Delta v_x}{\Delta t} \tag{4.4}
\]

The SI unit for acceleration is meters per second per second (m/s/s), more commonly expressed as meters per second squared (m/s2). Since velocity is a vector, acceleration must also be a vector. For two-dimensional motion, we use the rate of change of each component of the velocity:

\[
\vec{a} = \underbrace{\left(\frac{\Delta v_x}{\Delta t}\right)}_{a_x}\hat{x} + \underbrace{\left(\frac{\Delta v_y}{\Delta t}\right)}_{a_y}\hat{y} \tag{4.5}
\]

However, the direction of acceleration alone does not tell you anything about which direction you are traveling; it only tells you how the velocity is changing. If your acceleration is in the same direction as your velocity (whether that is positive or negative) you are “speeding up;” the magnitude of your velocity is increasing. Conversely, if your acceleration is in the opposite direction as your velocity, you are “slowing down.” (In physics we generally do not use the term “deceleration.”) This is summarized for one-dimensional motion in the table below.

Direction of \(\vec{v}\)Direction of \(\vec{a}\)Speed (\(v\)) is…
++Increasing
+Decreasing
+Decreasing
Increasing
Table summarizing the relationship between direction of velocity, direction of acceleration, and speed.

practice

Practice

Practice

4.4.1 Determining acceleration from a velocity vs time graph

Just as we could produce a velocity vs. time plot from position vs. time data, we can take our knowledge of how an object’s velocity is changing with time, and produce an acceleration vs. time graph. In the case of constant velocity (such as in the figures of the previous section), that graph would simply be a straight line at \(a = 0\).

You’ll notice that the mathematics involved here are exactly the same as what we used to create velocity graphs from position graphs! So, the section 4.3 example and practice problems will build the exact same skills as you need for this material. This is one of the beauties of mathematics; the same math can describe different physical situations.