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.
Equal
distances will be measured out along level ground by the class. This
is most likely to be done using the porch of the FSM-China Friendship
Center. There is a nice 30 meter stretch of porch that can be marked
off into five meter sections with the last timing mark at the west
wall.
The ball will be rolled at different speeds using an underhand bowling style of pitch along level ground at a variety of speeds. At each timing mark a timer will record the time at which the ball passes their timing mark. Recorders will record the times at all timing marks. The remaining students will pick up loose balls.
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.
This procedure may be modified on the day of the laboratory due to some timer issues.
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.
Ball runs:
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Roll one |
Roll two |
Roll three |
Roll four |
Roll five |
Roll six |
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y |
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Time/s |
Distance/m |
Time/s |
Time/s |
Time/sec |
Distance/m |
Time/s |
Distance/m |
Time/s |
Distance/m |
Time/s |
Distance/m |
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The data tables can be copied and pasted from the spreadsheet to the word processor. This will be done as a show and tell in class.
Pick two runs that took different lengths of time to get to 30m. Report the slope for the two ball rolls you've chosen using the function described below. The y-intercept, the starting distance, will be zero meters. The ball will always be rolled from the starting line of zero meters.
A spreadsheet is used to find the slope and the y-intercept of the best fit line through the data.
To get the slope m with OpenOffice Calc use the function:
=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.
An example where the data is in the first two columns from row three to nine can be seen below.
=LINEST(B3:B9;A3:A9;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 = slope (speed) × time
d = v × t
This sometimes written simply as d = vt
Given any time, we can calculate the distance. Given any distance, can solve for the time.
Microsoft Excel can be used to create a chart that displays the linear regression trendline. The formulas for 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)
Two XY scatter graphs are required – one for each ball run you chosen to use. The graphs 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.
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.