This laboratory explores the relationship between time and distance for an object moving at a constant acceleration. In this situation the velocity is changing.
What is the nature of the mathematical relationship between time and distance for a falling object?
Existing gravitational theory asserts that the distance an object falls when dropped is given by the mathematical equation:
or
.
A graph of time versus distance should result in the half-curve of the start of a quadratic parabola as seen on the right.
This graph suggests that time and distance are not related linearily. That is, twice as much fall time results NOT in twice as much distance fallen, but in MORE THAN twice as much distance fallen. Look at the graph – a fall of 0.4 seconds falls about 80 cm, but 0.8 seconds falls over 300 cm. When the fall time doubles, the distance appears to QUADRUPLE.
Confirming the hypothesis that a time versus distance graph is a quadratic curve is difficult. We cannot determine the slope of a curve using a best fit straight line. The slope would be in centimeters per second – speed, but the slope is changing, the line is curved, which means the speed must of the falling object must be changing.
If, however, we square the time values and then graph these squared time values on the x-axis and the distance values on the y-axis, then a straight line with a slope of one half g should be the result:
This is just like y = mx except that for x we are going to graph the square of the time (t²). If all goes well, this second graph should be a straight line. The values on your axes will differ from those seen here.
The units of slope for the second graph and of gravity in this laboratory are centimeters per second squared, also written cm/s².
Note that your graphs based on your data from laboratory might not produce lines as smooth as those seen above. Small deviations from a smooth line are the result of small errors in measurement, not evidence that the theory is false. The whole pattern of the data would have to disagree with shape proposed to disconfirm the theory.
Teams will be formed, probably no more than six. Each team will have a ball, a stopwatch, and three meter sticks. Teams should divide themselves up into ball dropper/timers, meter stick holders, and recorders.
Small balls will be used. The ball will be dropped five times from each height and the time to fall will timed with the stopwatch. This will be repeated at each drop height to get the mean fall time for each drop height. The mean fall time is being used to improve precision. For the highest drops the tallest team member might be necessary. Be careful. Dropping a team member in lieu of a ball does not count!
Data will be recorded into a table and then plotted on graph paper, using the mean time in seconds on the horizontal x axis and the drop height in centimeters on the vertical y axis.
For data analysis a second table will be prepared using the square of the time in seconds versus the drop distance. This data will also be plotted on a graph sheet.
| Graph these two columns: If the theory holds, then the result is a parabola |
||||||
|---|---|---|---|---|---|---|
| Drop time trials/s | x | y | ||||
| Drop one | Drop two | Drop three | Drop four | Drop five | Mean drop time x (s) | Distance y (cm) |
| 000 | ||||||
| 050 | ||||||
| 100 | ||||||
| 150 | ||||||
| 200 | ||||||
| 250 | ||||||
| 300 | ||||||
Graph the x versus y data for the above table but do not try to calculate the slope. If the theory is correct, then the graph should be a gentle curve with a parabolic shape. Remember – includes units in the header cells of the table. Do not put units in the data cells of the table in a spreadsheet. The "letters" will cause a spreadsheet to fail to graph the data as xy scattergraph data.
Use your calculator to square the mean drop times in the table above and record the results below.
| Graph these two columns. if the theory holds, then this data should plot as a straight line | |
|---|---|
| mean drop time² x² (s²) put the square of the mean drop time values in this column! |
distance y (cm) |
| 000 | |
| 050 | |
| 100 | |
| 150 | |
| 200 | |
| 250 | |
| 300 | |
Plot the data in this table. Your points should be roughly along a straight line. Be careful when plotting. Make sure your axes are laid out with scales where equal distances are equal changes in values along that scale.
Some calculators can perform a linear regression. Your instructor will assist groups with determining the slope and intercept for their data using their calculators.
Using a calculator determine the slope of the line for the time squared versus distance data. Note that we are calculating the slope m for the quadratic equation y = mx² because our data table is using the square of the x-values.
Thus the slope m [a] is equal to half of the acceleration of gravity g [a]. We can multiply our slope by two to calculate the acceleration of gravity for our ball drop.
therefore
On the second graph the rise is centimeters and the run is seconds². Slope is rise over run. Therefore the units of slope and of the acceleration of gravity are both cm/s².
Use your g and the formula:
to predict the fall time t for a drop d of 500 centimeters.
The "textbook" value for the acceleration of gravity g at earth's surface is 980 cm/s². How close did you come to this result? Calculate (your value of g − 980)/980 to determine the percentage difference between your g and the value quoted in science texts. [a]
If you use Excel to calculate your slope [a] you can force the intercept [a] to be zero by using:
=LINEST(distance (y) values,time² (x²) values,0)
Note that this slope will not agree exactly with the calculator slope as the calculator cannot be forced to set the intercept to zero.
For data display two graphs are required. The first graph will depict time versus distance, the second will plot the time squared against the distance. Use a spreadsheet to generate these graphs and then copy and paste the graphs into a word processor for your report.
Discuss the nature of the mathematical relationship between time and distance for a falling object. Discuss whether the first graph could be parabolic, allowing for the uncertainty the small errors that may cause the points to "wiggle" slightly. Discuss whether the second graph is a line. Report the slope m and the calculation for the acceleration of gravity g. You can compare your result to the textbook result of 980 cm/s² Discuss any problems you encountered in this laboratory including those that may have contributed to uncertainty in your measurements.