John P. Crotty
When we perform calculations, we must consider the accuracy of the numbers we are using. The accuracy of our results is determined by the accuracy of these numbers. The numbers that we used in our experiments were approximate numbers, since we used a measuring process to obtain them.
When we measured our course, we marked off ten yard intervals using a measuring tape. The ten yards were approximate. The tape may not be calibrated exactly or more likely may have stretched with usage. A more accurate measuring device may have given us different readings.
Temperature and humidity also affect measurement. Look at the difference in the sag of telephone wires between the winter and the summer. The lines are stretched tight in the winter, but droop in the summer. In the winter, the basketball that bounced fine in the gym appears to be fIat when you take it outside. I see it in industry, where the factory conditions can affect a material to such an extent that the precision of the design is lost. An example of this would be the O-rings on the space shuttle.
The human factor also has to be considered. Today’s stopwatches measure to hundredths of a second. Did my timers all start their watches at the same time? My son is a swimmer. At a swim meet, there are three timers in a lane. In most races there is a range of at least fifteen-hundredths of a second between the three watches. But, go back before the electronic stopwatches and stopwatches were only calibrated to a tenth of a second.
When we write a number, we have to be aware of how many of its digits are significant. All nonzero digits are significant. A trailing zero may be significant or it may be a placeholder. The accuracy of a number is determined by the number of significant digits it contains. The last significant digit of an approximate number is not completely accurate. It has usually been found by estimating or rounding off.
When we work with approximate numbers, we have to be careful about how we write our answers. Especially when using computers, the temptation is to express our answers with too many digits. Precision is related to accuracy. The precision of a number refers to the decimal position of its least significant digit.
When we work with approximate numbers, we should remember these rules:
-
1. When approximate numbers are added or subtracted, the result is expressed with the precision of the least precise number.
-
2. When approximate numbers are multiplied or divided, the result is expressed with the accuracy of the least accurate number.
It is important to remember the above. It is too easy to fall into the habit of thinking that the answer is accurate or precise to an indefinite degree because the computer did it. An even worse mistake though is to think that an answer is correct just because it was done on a computer.
David Coasting Down a Slight Incline
(Distance vs. Time)
____
____
Riding Position
____
____
____
____
Wearing
Distance
|
Normal
|
Streamline
|
Narrow
|
Coat
|
(yds)
|
(sec)
|
(sec)
|
(sec)
|
(sec)
|
0
|
0.00
|
0.00
|
0.00
|
0.00
|
10
|
4.49
|
4.27
|
3.77
|
4.81
|
20
|
8.79
|
6.40
|
5.77
|
7.17
|
30
|
8.74
|
8.19
|
7.43
|
9.22
|
60
|
13.16
|
12.44
|
11.42
|
13.90
|
90
|
17.33
|
16.47
|
15.09
|
18.21
|
120
|
21.39
|
20.29
|
18.63
|
22.31
|
David Coasting Down a Slight Incline
(Distance vs Average Velocity)
____
____
Riding Position
____
____
____
____
Wearing
Distance
|
Normal
|
Streamline
|
Narrow
|
Coat
|
(yds)
|
(ft/sec)
|
(ft/sec)
|
(ft/sec)
|
(ft/sec)
|
0
|
0.0
|
0.0
|
0.0
|
0.0
|
10
|
6.7
|
7.0
|
8.0
|
6.2
|
20
|
8.8
|
9.4
|
10.4
|
8.4
|
30
|
10.3
|
11.0
|
12.1
|
9.8
|
60
|
13.7
|
14.5
|
15.8
|
12.9
|
90
|
15.6
|
16.4
|
17.9
|
14.8
|
120
|
16.8
|
17.7
|
19.3
|
16.1
|
David Coasting Down a Slight Incline
(Distance vs Instantaneous Velocity)
____
____
Riding Position
____
____
____
____
Wearing
Distance
|
Normal
|
Streamline
|
Narrow
|
Coat
|
(yds)
|
ft/sec)
|
(ft/sec)
|
(ft/sec)
|
(ft/sec)
|
0
|
0.0
|
0.0
|
0.0
|
0.0
|
10
|
6.7
|
7.0
|
8.0
|
6.2
|
20
|
13.0
|
14.1
|
15.0
|
12.7
|
30
|
15.4
|
16.8
|
18.1
|
14.6
|
60
|
20.4
|
21.2
|
22.6
|
19.2
|
90
|
21.6
|
22.3
|
24.5
|
20.9
|
120
|
22.2
|
23.6
|
25.4
|
22.0
|
David Coasting Down a Slight Incline
(Distance vs Acceleration)
____
____
Riding Position
____
____
____
____
Wearing
Distance
|
Normal
|
Streamline
|
Narrow
|
Coat
|
(yds)
|
(ft/sec/sec)
|
(ft/sec/sec)
|
(ft/sec/sec)
|
(ft/sec/sec)
|
0
|
0.0
|
0.0
|
0.0
|
0.0
|
10
|
1.5
|
1.6
|
2.1
|
1.3
|
20
|
2.8
|
3.3
|
3.5
|
2.7
|
30
|
1.2
|
1.5
|
1.9
|
0.9
|
60
|
1.1
|
1.0
|
1.1
|
1.0
|
90
|
0.3
|
0.3
|
0.5
|
0.4
|
120
|
0.1
|
0.3
|
0.3
|
0.3
|
Kasey Pedaling
(Distance vs Time)
|
|
Velocity
|
Velocity
|
|
|
Computed
|
from
|
Distance
|
Time
|
v=d/t
|
tangent
|
(yds)
|
(sec)
|
(ft/sec)
|
(ft/sec)
|
0
|
0
|
0
|
0
|
10
|
3.52
|
8.5
|
10.7
|
20
|
5.67
|
14.0
|
17.1
|
30
|
6.99
|
22.7
|
21
|
60
|
10.05
|
29.4
|
30.5
|
90
|
13
|
30.5
|
32.1
|
120
|
15.87
|
31.4
|
did not draw
|
Kasey Pedaling
(Velocity vs Time)
____
____
Distance
____
____
from
|
|
Area
|
Actual
|
Velocity
|
Time
|
A=(bl+b2)h/2
|
Distance
|
(ft/sec)
|
(sec)
|
(ft)
|
(ft)
|
0
|
0
|
0
|
0
|
8.5
|
3.52
|
15
|
30
|
14.0
|
5.67
|
24
|
30
|
22.7
|
6.99
|
24
|
30
|
29.4
|
10.05
|
80
|
90
|
30.5
|
13
|
88
|
90
|
31.4
|
15.87
|
89
|
90
|
(figure available in print form)
(figure available in print form)
(figure available in print form)
(figure available in print form)
(figure available in print form)
(figure available in print form)
(figure available in print form)