To produce a velocity that can be reproduced a ball will be rolled down a ramp. Some goals
This laboratory explores the relationship between time and distance for an object moving at a constant velocity. In physical science a "relationship" means how one variable changes with respect to another variable. This change is described using mathematical equations. Math is the language in which physics is "spoken."
For a physicist mathematics is not just a tool by means of which phenomena can be calculated, it is the main source of concepts and principles by means of which new theories can be created... ...equations are quite miraculous in a certain way. I mean, the fact that nature talks mathematics, I find it miraculous. I mean, I spent my early days calculating very, very precisely how electrons ought to behave. Well, then somebody went into the laboratory and the electron knew the answer. The electron somehow knew it had to resonate at that frequency which I calculated. So that, to me, is something at the basic level we don't understand. Why is nature mathematical? But there's no doubt it's true. And, of course, that was the basis of Einstein's faith. I mean, Einstein talked that mathematical language and found out that nature obeyed his equations, too. – Physicist Freeman Dyson
What is the nature of the mathematical relationship between time and distance for an object moving a constant velocity?
How does this relationship change with changes in speed?
For a rolling ball, distance increases as time the ball rolls increases. For a ball rolling at a constant velocity (speed), this relationship is predicted to be linear. That is, a graph of time versus distance should generate a straight line. The equation that is proposed is that distance = velocity * time. If time is graphed on the x-axis and distance on the y-axis, then the slope of a line through the data should be the velocity. If a graph does not produce a line, then the hypothesis of a linear relation between time and distance is disproved for a rolling ball.
Stopwatch details: Press B until you see stopwatch mode. The stopwatch also has a regular time mode and a set time mode. Press C to start and stop the timer. After stopping, press A to reset the timer.
To produce a velocity that can be reproduced a ball will be rolled down a ramp.
Students with stopwatches will be assigned to keep track of where the ball is at one, two three, four, five and six seconds. Do not try to stop the timer - just watch for the second to which you are assigned. Use your zori to mark where the ball was at that time.
The distance in centimeters to each timer will be recorded in the table. The experiment will be repeated and the average distance to a particular time point will be calculated.
The whole procedure will be repeated for a different ball speed to determine whether different speeds produce different slopes.
| time (s) | distance trial one (cm) | distance trial two (cm) | distance trial three (cm) | Average fast dist (cm) |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 1 | ||||
| 2 | ||||
| 3 | ||||
| 4 | ||||
| 5 | ||||
| 6 |
Note: the average distance can be calculated in the computer laboratory during the second portion of the laboratory.
Write the data for a single run of a slower ball (half-way up the ramp) in the third column below. Copy the fast ball averages (calculated in the computer laboratory) into the middle column below)
| time (s) x | avg fast dist (cm) (copy from above) y1 |
slow dist (cm) y2 |
|---|---|---|
| 0 | 0 | 0 |
| 1 | ||
| 2 | ||
| 3 | ||
| 4 | ||
| 5 | ||
| 6 |
After gathering the data, the class will go to the A204 math/science computer laboratory to determine the relationship, if any, between the time and distance.
To calculate the mean distance in the right column of the first table, use the average function.
=AVERAGE(data)
A graph is one of the first tools one uses when exploring new data. A graph will visually indicate whether a two variables are related and and what that relationship might be. Using the second table above, make a single xy scatter graph with both the fast and slow ball data. Instructions for making graphs using OpenOffice.org Calc or Microsoft Excel are also available on line.
The graph can be produced in OpenOffice.org Calc or in Microsoft Excel and then copied to OpenOffice.org Writer or Microsoft Word. Click on the graph, press ctrl-c to copy, switch to Writer or Word, and press ctrl-v to paste the graph into your document.
A spreadsheet function is used to find the slope of the best fit line through the data. Note that the ball started at a time and distance of 0 seconds at 0 centimeters. The y-intercept is (0 s, 0 m). A function called the LINear Estimator function is used to calculate the slope with a y-intercept forced to be equal to zero.
To get the slope m with OpenOffice Calc use the function:
=LINEST(distance (y) values;time (x) values;0)
The formulas for Microsoft Excel are almost the same as for OpenOffice Calc with one difference: Excel uses a comma instead of a semi-colon.
=LINEST(distance (y) values,time (x) values,0)
Note that the y-values are entered first, the x-values are entered second. This is the reverse of traditional algebraic order where coordinate pairs are listed in the order (x, y). The x and y-values are usually arranged in columns. The column containing the x data is usually to the left of the column containing the y-values.
The final zero in the formula forces the spreadsheet to make the y-intercept, the starting value, be zero meters at a time of zero seconds.
An example where the data is in the first two columns from row two to seven can be seen below (row one contains the column head names).
=LINEST(B2:B7;A2:A7;0)
In algebra the equation of a line with a zero intercept is written as y = mx where m is the slope. In physical science the slope is usually a rate of change. In this laboratory the slope is the speed of the ball. We use the letter v for velocity (speed in a direction) in formulas.
distance d = slope (speed v) × time t
d = v ×t
This sometimes written simply as d = vt
Given any time t, we can calculate the distance d. Given any distance d, can solve for the time t.
Discuss the nature of the mathematical relationship between distance and time for a rolling ball. Discuss how different starting speeds affect the slope of the relationship. Discuss whether the relationship appears to form the straight line predicted by the d = vt theory.