AP® Physics 1: Algebra-Based · Unit 1 · Topic 1.3
Lab Skill: Linearizing Motion Data
Linearizing means re-plotting curved motion data so it forms a straight line, because a straight line has one measurable slope you can turn directly into a physical constant.
Position versus time for a falling or accelerating object is a curve, a parabola, because position depends on t2. Curves are hard to measure: you can eyeball a smooth line through the points, but "how curvy is it" is not a number. So physicists transform the data instead. For constant acceleration starting from rest, position = (1/2)a t2. If you plot position on the vertical axis against t2 on the horizontal axis, instead of t, the relationship becomes position = (a/2) x (t2), which has the form y = (slope) x, a straight line through the origin.
Once the data is a straight line, you draw the best-fit line and measure its slope using rise over run, exactly like any linear graph. That slope is not the final answer by itself; it is (a/2), or (g/2) for a falling object, because of how the equation was built. To recover the constant you actually want, you undo the algebra: multiply the slope by 2. A slope of 4.9 means g/2 = 4.9, so g = 9.8 m/s2.
This lab skill sits right after you have learned to read the three motion graphs. Position-time, velocity-time, and acceleration-time graphs describe motion once you already know its shape. Linearizing is what you do with real, messy data before you know the shape for certain: it is how you test whether acceleration really is constant, and how you pull a trustworthy number like g out of an experiment rather than assuming it.
Key terms
- Linearize
- To replot data using a transformed variable, like t2, so a curved relationship becomes a straight line.
- Slope
- The rise over run of a straight-line graph, which here equals half the acceleration.
- Best-fit line
- The straight line drawn through scattered data points that best represents the overall trend.
- Free fall
- Motion under gravity alone, with constant downward acceleration g equal to 9.8 m/s2 near Earth's surface.
Watch it explained
- How it worksWalk me through plotting position against time squared to straighten out free-fall data.
- How it worksHow does the slope of a linearized graph hand you the physical constant you were after?
- Worked exampleFind the value of g from a position versus time-squared graph whose slope is 4.9.
- Common mistakeWhy isn't sketching a smooth curve through the points as useful as linearizing them?
Check yourself
End of topic.