Modern Galileo Experiment
In Galileo’s Dialogues Concerning Two New Sciences, he notes that a ball rolling down a uniform incline accelerates uniformly. Unfortunately, the most accurate tool to available to Galileo was a water clock, and this experiment has been designed to confirm his results using more-precise measurements from a motion detector and computer software. Similar to previous experiments, a test-cart was observed in motion down an incline track and using the Logger Pro software, precise graphical representations were created.
Data Table for Cart on Incline:
| Data Point | Time (s) | Speed (m/s) | Change in Speed (m/s) |
|
1 |
1.4 |
0.371 |
0 |
|
2 |
1.5 |
0.397 |
0.026 |
|
3 |
1.6 |
0.424 |
0.027 |
|
4 |
1.7 |
0.452 |
0.028 |
|
5 |
1.8 |
0.478 |
0.026 |
|
6 |
1.9 |
0.502 |
0.024 |
|
7 |
2 |
0.529 |
0.027 |
|
8 |
2.1 |
0.554 |
0.025 |
|
9 |
2.2 |
0.579 |
0.025 |
|
10 |
2.3 |
0.605 |
0.026 |
| slope of v/t graph: 0.258 m/s2 | |||
| average acceleration: 0.26 m/s2 | |||
Analysis:
2. As stated earlier, Galileo’s definition of uniform acceleration is equal increases in speed in equal intervals of time. Do your data support or refute this definition for the motion of an object on an incline? Explain.
The data recorded in this experiment supports Galileo’s definition of uniform acceleration. Analysis of the data revealed an average change in speed of 0.26m/s2, and that the largest variation from this average in any data point (change in speed) at a mere +/- 0.002 m/s. This follows that there were equal increases in speed in equal intervals of time.
3. Was Galileo’s assumption of constant acceleration for motion down an incline valid? How do your data support your answer?
Galileo’s assumption of constant acceleration for motion down an incline was quite valid. The data recorded shows that the slope of v/t graph is equal to 0.258 m/s2, while the average acceleration is calculated to be 0.26 m/s2. The uniformly changing velocity observed in the cart’s motion supports Galileo’s conclusions.
5. Compare the slope of the v vs t graph and the average acceleration. Why are these values not exactly the same?
These values differ by 0.002 m/s2, and the reason they are not the same is that the average acceleration was calculated using only two data points, while the slope of v/t was determined using the entire range of motion.
6. Draw a graph that represents the motion of a car whose initial velocity is greater than zero and which has a positive acceleration (vf >vo). Using the correct variables, write an equation for the area under the curve and the slope. Solve for vf in the slope equation and substitute into the area equation. Simplify and set equal to zero. Arrange in order of ascending powers of t.
7. Look at the curve fit equation for the position vs. time graph. What does the constant c represent? How does the constant c relate to the slope of the velocity vs. time graph?
The constant C represents the coefficient value for the x2 value of the position vs. time graph. This coefficient is equal to 0.129, and the derivative of the Cx2 is related to the slope of velocity vs. time, where slope is ½ (C), or 0.258.
8. Look at the curve fit equation for the velocity vs. time graph. Does the fitted function have a constant slope? What does the slope represent? What are its units? Record the slope in the data table.
The fitted function has a constant slope of 0.258, which represents the acceleration of the cart, in units m/s2.
Questions:
1. What would happen if you simultaneously dropped the two balls, but this time held the small ball about 30 cm above the larger one. Would the distance between the two balls increase, decrease, or remain the same as they fall? Since the fall time is short, it is hard to tell just what happens by eye. You will see why Galileo and the people of his day had a difficult time answering the questions of motion. Try to answer this question using Galileo’s assumption of constant acceleration.
The distance between the two balls would remain the same as they fall because they are both experiencing equal changes in speed in equal periods of time.
Conclusion:
The above observations and analysis conducted in this experiment has shown agreement with Galileo’s theories. Although he possessed primitive tools, his assumptions were correct, and use of modern technology has validated this.
