Linearizing Graphs: Drop Height vs Time

Scenario

Angela, Blake, and Carlos have been given a stopwatch, several large spheres, and a meterstick and have been asked to determine the acceleration due to gravity. They were told that they need to collect drop height and time data at several different heights to create a position vs. time graph. The averages of the collected data are shown in the data table below.

Part A: Graph the drop height with area or ball on the axis above.

Part B: Based on your graph and the table at the right, identify the correct relationship between the drop height and the time to fall to the ground.

Claim: The displacement is proportional to the square of time.

Part C: The relationship between drop height and time to fall can be compared to the equation H = ½gt², so that the students can create what is called a linearized graph. Fill in the third column in the data table with appropriate values and graph to create a linearized graph.

Part D: What quantities should be plotted on a graph if the graph is to have a linear trend and the slope of the best-fit line is to be used to determine the acceleration due to gravity?

Acceptable answers include: Drop height vs. time squared, or the square root of drop height vs. time, among others.

Drop Simulation

0.50 m

Collected Data

Height H (m) Time T (s) T² (s²)
No data collected yet. Drop the ball to collect data!

H vs T (Original)

H vs T² (Transformed)

Common Relationships Used in AP Physics 1

Graph Relationship
Linear As x increases, y increases proportionally. y is directly proportional to x.
Inverse As x increases, y decreases. y is inversely proportional to x.
Quadratic y is proportional to the square of x.
Square Root The square of y is proportional to x.

Understanding Linearization

Not all relationships are linear, but when you manipulate the data so that the graph is a line, it is easier to get usable information from the graph to be able to draw conclusions as well as construct equations.

In this experiment, the relationship between drop height (H) and time (T) follows the equation:

H = ½gT²

This is a quadratic relationship - the height is proportional to the square of time. To linearize it, you need to find the right transformation.

Try different transformations:

  • T² (square): Creates a linear graph where slope = ½g
  • 1/T (inverse): Creates a curve (not linear)
  • √T (square root): Creates a curve (not linear)
  • T (no transformation): Shows the original quadratic relationship

When you find the correct transformation (T²), the slope of the line equals ½g, so g = 2 × slope. The R² value tells you how linear the fit is - values close to 1.0 indicate an excellent linear relationship!