AP Physics 1

Linearisation of Data

The purpose of most physics experiments is to determine relationships between physical variables in an observed phenomenon. Reliable relationships expressed in a formula involving the variables allows one to predict a physical outcome when there is a change in another physical variable.

Linearisation of Experimential Data

Given the following Data we wish to understand the relationship between Distance and Time

Time (s) Distance (m)
0.01 0.00
0.10 0.05
0.20 0.11
0.30 0.43
0.40 0.75
0.50 1.14
0.60 1.84
0.70 2.53
0.80 3.45
0.90 3.67
1.00 4.87

Step 1) Plot the data

Plot the Data Using Computer Software

Ball Falling Data
Plot of Ball Falling when released from rest
Step 2) Identify the General Relationship

Given Our Data Plot and the Plots of our Basic Relationships We need to choose the one that is the best match when compared to our data

Plot Basic Relationship Our Data Plot
Linear Plot
Linear Plot
y = kx
Ball Falling Data
Plot of Ball Falling when released from rest
Quadratic Plot
Quadratic Plot
y = k x2
Square Root Plot
Square Root Plot
y = sqrt(x)
Inverse Plot
Inverse Plot
Inverse Square Plot
Inverse Square Plot
y = 1/x2

In Our Case our data is best matched by the Quadratic Plot y = k x2

Step 3) Linearise the Data

Determine How we Need to Modify Our Data to get a Linear Plot

So as we can see we will need to square the time and then create a new plot

Step 4) Plot the Linearised Data

Data after Modification

Time (s) Time2 (s2) Distance (m)
0.01 0.0001 0.00
0.10 0.010 0.05
0.20 0.04 0.11
0.30 0.09 0.43
0.40 0.16 0.75
0.50 0.25 1.14
0.60 0.36 1.84
0.70 0.49 2.53
0.80 0.64 3.45
0.90 0.81 3.67
1.00 1.00 4.87

We will plot Distance (y axis) versus Time2 (x axis)

Ball Falling Data Linear
Linearised Plot of Our Falling Ball
Step 5) Draw your Conslusions

Understanding Our Results

Slope of Our Line 4.898 m/s2

Y - Intercept effectivly zero

Compare to Actual Physics Equations

Equation that Governs a Falling Ball

Distance = (1/2)ag time2

This means that our slope is equal to (1/2) ag, where ag is the acceleration due to gravity

4.898 = (1/2) ag

ag = 9.79 m/s2 Which is our measured acceleration due to gravity

The accepted value for the acceleration due to gravity on Earth is 9.81 m/s2 so we
are in very good aggrement.