Various water levels were measured and timed to determine the length of time each different height took until the water had poured out of the can. A graph was made and a linear regression line was plotted and determined; the equation was t = (3�1)h + (20�7). The dependent variable was time because it had to be observed while the independent variable, the water level, was observed. As well, it was determined that if the length of the hole was 3 times longer, the time for the water to pour out of the can would take 87.5 – 90% less. Furthermore, if this experiment was done on the moon, assuming that flow rate is constant, the water would pour down 40.6 – 41% slower than if it was on Earth.

Theory:

Arithmetic mean

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[1]

Where, x = arithmetic mean

xi= the ith term

n = number of terms

Standard Deviation

[2]

Where, ? = standard deviation

xi = the ith value

x = the arithmetic mean of values

n = number of values

Uncertainty Addition

For (z ? ?z)

z = x1 + x2 + … + (xn)

[3]

Uncertainty Multiplication/Division

The multiplication and division uncertainties were calculated using the following formula,

or [4]

Uncertainty Exponential

To calculate the uncertainties for equations using exponents, the following formula was used,

[5]

Where, for [3], [4], and [5],

n = exponent

z = mantissa variable

x = mantissa variable

y = mantissa variable

?z = uncertainty of z

?x = uncertainty of x

?y = uncertainty of y

Bernoulli’s Principle

The mean outflow speed of water can be calculated using an application of Bernoulli’s principle. The following equation is an altered form of it so that time can be solved,

[6]

Where,

t = Time it takes for the water to flow out of the can (s)

g = Acceleration due to gravity (ms-2)

h = Water level (m)

Rectangle Volume

The volume of any rectangle can be determined using the following formula,

V [7]

Where,

V = Volume (cm-3 or mL)

x = Width (cm)

l = Length (cm)

z = Height (cm)

Flow Rate of Water

The flow rate of water is calculated using the following formula,

[8]

Where,

V= Flow rate (mLs-1)

Vol = Volume (mL)

t = time (s)

Gravitational Constant of Earth’s Acceleration

The gravitation constant of Earth’s acceleration, with an implied uncertainty, that was used for this experiment was (9.81�.01) ms-2.

Method:

The tin can, ruler, stopwatch, and tape were gathered. A piece of measuring tape was taken and increments of 1 cm were marked using the ruler. The tape was then stuck to the inside of the can carefully. The hole of the tin can was measured and plugged, using a finger. The time was recorded by counting down from three seconds and removing the finger. Once five drops were counted coming out of the bottom of the can, the timer was stopped and the results were recorded. Twelve different water levels were recorded, below 3/4 of the can’s height. For at least one of the measurements, the time was measured another nine times. As well, the time was recorded between the fourth and fifth heights and when the can was full.

Data

Time at 6cm

(s)

1.

48.22

2.

29.38

3.

50.03

4.

50.84

5.

49

6.

48.91

7.

30.4

8.

49.6

9.

48.56

10.

46.63

Mean

45.157

?

8.12712

Water Level (h)

(�.02)cm

Time (t)

(�8)s

1

13.81

2

18.85

4

38.16

5

44.22

6

48.22

7

53.18

8

45.38

10

68.07

11

38.5

12

54.12

Analysis

1. See figures [2.0] and [3.0].

2.

a. Predicting the time using the graph,

i. 5.5 cm – 46:00 seconds

ii. Full (17.3 cm) – 85:00 seconds

b. Checking prediction experimentally,

i. 5.5 cm – 30.75 seconds

ii. Full (17.3 cm) – 55:88 seconds

3. See figure [1.0]. This value is higher than norm because of the formation of vortexes. This is explained in the errors.

4. The independent variable is the water level, while the dependent variable is time. This is because the independent variable is the one that’s being changed so that the dependent variable can be observed. In this case, the water level is being manipulated, and the time it takes to empty the can is being observed.

5. The straight regression line that was obtained has the equation t = (3�1)h + (20�7).

6. The units for the constant of proportionality would have to be s cm-1.

7. Using the equation in question 5, the times for the water levels of 5.5 cm and 17.3 cm are (37�9) seconds and (70�20) seconds, respectively. Using the regression equation, the times were much more accurately predicted.

8. Both questions apply uncertainty formulas from [3], [4], and [5]. Also, they’re assuming that the flow of water is constant.

a. Using [7], the volume of rectangular shaped water block with a height of 5 cm and base of 1cm-2 is determined. Using [8], and an arbitrary value for time (;1), the flow rate is found. Similarly, using [6], the flow rate was found for three times the length, using the same time. Comparing the two, the ratio is found: (9�1). Therefore, if the length is 3 times longer, it would only take approximately 10 – 12.5% of the time it would for the original dimensions to flow down.

b. According to equation [6], the time it takes for the water to flow out was determined – the times for Earth and the Moon, using an arbitrary height (;1) and their respective accelerations of (9.81�.01) ms-2 and (1.6�.2) ms-2, were found: (0.226�.001) seconds and (0.5530�.0005) seconds respectively. The time it would take to flow out of the can, the ratio of the moon to earth, was found to be (0.408�.002). Therefore, it will flow down approximately 40.6 – 41.0% slower on the moon.

Errors

The temperature of the water was not kept constant. When hot water was used, a vortex formed when the water was draining; this increased the time it took the drain the water out of the can. However, when cold water was used, there was no vortex, so the time interval was shorter. The observers were unaware that this would occur at the beginning of the experiment.

To minimize the errors, the tape was used so that the ruler would not have to be dipped inside the water-filled can each time. This prevented the ruler from only affecting the water level. As well, to ensure that the tin can was as stable as possible, the can was placed near the edge of the sink. By remaining stable, the wobbling factor’s result would be minimized.

Conclusion

The independent variable is the water level, while the dependent variable is time. This was decided upon because the water level is being manipulated so that the different times can be observed. By plotting the data points and using this relationship, the linear regression equation was found to be t = (3�1)h + (20�7). As well, if this experiment was done on the moon, assuming that flow rate is constant, the water would pour down 40.6 – 41% slower than if it was on Earth. Furthermore, the ratio between the larger hole and original hole was (9�1).

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