Annie and On: 2004
Each year, the student body from the International School Bangkok are required to participate in a field trip known as Week Without Walls. This trip provides opportunities for students to travel the world, to come into contact with the beliefs and practices of various cultures from different countries. A recent group visited Kunming, China. The trip included a visit to one of Kunming's traditional tea centers, where a ceramic doll was presented as a tool for measuring whether the water had been heated to a suitable temperature for brewing tea. The instructor claimed that if the water that was poured on to the doll is at the suitable temperature, then the doll would project water, "pee", in a smooth path; and the projected water will travel the distance of (approximately) 1 to 1.5 meters.
The doll demonstrates the relationship between pressure and temperature change. It is first be submerged in hot water. The hot water increases the air pressure inside, forcing some air to move out of the doll. The doll is then submerged in cold water, the air inside contracts and water is forced in. If hot water is now poured on to the doll which is half full of water, the remaining air inside will expand causing the doll to "pee".
Projectile Motion
Projectile motion is movement with constant acceleration in one specific direction described with the following equation....
... where s is the distance traveled; u is the initial velocity; t is the time; and a is the acceleration. s, u and a are vectors. The equation written in column vector form becomes....
Since this is a quadratic equation, it shows that the motion of a projectile follow a parabolic path.
Research question
What is the relationship between the temperature of
the water poured onto the "pee-man doll" and the pressure
inside the doll.
Hypothesis
The hotter the water poured onto the doll, the further the water will project from the doll. There is a linear relationship between the temperature of the water poured over the doll and the pressure inside the doll.
Explanation
When hot water is poured over the doll, the temperature of the doll and the air inside will rise. The change in temperature will increase the pressure inside. The distance which the projected water will travel depends on the ejection velocity. Higher temperatures will increase this velocity.
PART 1: the relationship between ejection velocity and pressure
In this part of the experiment the relationship between velocity of a water jet (in a parabolic path) and the water pressure was investigated. A cylindrical bottle was selected for this experiment. The bottle was partly transparent so that the water level inside the bottle could be seen from the outside. A drill-press was used to drill a small hole on the side of the bottle. The distance from the bottom of the bottle to the hole was measured and recorded.
The bottle was filled with water unit the water leveled with the bottom of the hole (just so that the water would not leak out). The bottle was then placed on a scale to measure its mass. Scotch tape was placed over the hole, and water was added. In the first trial, the water level was a few centimeters higher than the hole. A ruler was used to measure the depth from the water surface to the hole. Then the bottle was placed on a scale to measure its mass. After the depth and mass were recorded, the scotch tape was removed and water was allowed to project from the hole. A ruler is used to measure the horizontal distance which the water had traveled R, from the bottom of bottle, directly underneath the hole, to where the water hit the floor.
The properties of a parabolic curve were used to calculate the ejection velocity.
At once ...
Using this equation, the horizontal velocity was found without knowing the time of flight.
PART 2: the peeing doll
The doll was half filled with cold water and a beaker was filled with hot water. The temperature of the water in the beaker was measured and recorded. Once the doll was in place hot water was poured over the doll. A digital VDO camera was used to record this experiment, placed as shown in the figure above. Black construction paper was used to make the water jet show more clearly when recorded on the digital camera. A meter stick was placed parallel with the background.
Afterward, these video images were upload to a Macintosh computer. The clips were shown on the computer, and the scenes where the water path was most clearly defined were selected (for each temperature). Selected frames were opened in Adobe Photoshop. The images were 'equalized' to make the water path appear more clearly, then a parabolic curve was fitted to the water path.
Note: the curve was made in the Graphical Analysis 2.0. This graph was copied and pasted to AppleWorks. Using the tool "ungroup picture", the parabolic curve was separated from the background and colored red. The parabola was then pasted to AppleWorks and superimposed on the path like this.
The equalized images were copied and pasted to AppleWorks. The red curve was adjusted and duplicated so that it fitted the water jet path. The "group" tool was used to group each image with its parabolic curve.
These images were printed out. The parabolas' maximum height and range were measured used as the data for analysis.
Data: PART 1
The depth of water above the hole in the cylindrical bottle with a diameter of 7.4 cm.
Data: PART 2
A frame was extracted from the movie as a JPEG file. The image was equalized in Adobe to make the water path more clearly defined. A parabola was then fitted to the path. The maximum height and half the range was estimated from the final image to within ± 2 mm on the printed photographs.
Table 1
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Note: The values listed above are the measurements taken from the photographs. The measurements were converted to actual heights and ranges using a scale factor determined from each photograph.
Analysis: PART 1
Data is listed in Table 2 below. Possible maximum and minimum value are listed to allow a calculation with errors.
The velocity of water as it leaves the bottle can be calculated using the calculation tools in Graphical Analysis. The function used was:
Table 2
This table shows the calculated water velocity as it leaves
the bottle (with errors).
Graph 1 shows that there is a direct relationship between the velocity of water as it leave the bottle and the depth of water. The greater is the depth, the faster the water will travel.
Table 3 This table shows the calculation of pressure (in Pascal) with errors.
Graph 2 showing the relationship between the velocity of water as it leaves the bottle and the pressure in Pascals. A straight line has been fitted to the data points as a first approximation.
The greater the pressure, the faster the water is projected. The approximation of this relationship is described by the equation ...
... where y represents the pressure (Pa) and x represents the velocity (cm/s).
Analysis: PART 2
The maximum heights and the relative ranges of the parabolic water paths are listed in the data section above, with errors of plus or minus 2 mm.
Note: The measurements of the Maximum Height and Range are measurements of the parabolic curve on the image, not the actual height and range of the water path.
The velocities were calculated as in Part 1 and the pressures inside the doll were estimated using the approximate pressure/velocity relationship. The estimates are listed in Table 4.
Table 4
Graph 3 shows the relationship between pressure and the velocity of the water as it leaves the container. The highlighted data points are from Part 1 of this experiment. These data points are used to create a function which approximates the relationship between pressure and velocity. The three pairs of non-highlighted data points are from Part 2 of this experiment. Their pressure values are determined from the approximation function.
Graph 4 shows the relationship between the pressure inside the doll and the temperature of the water poured over it.
Discussion
The data supports the hypothesis. The higher the temperature of the water that is poured over the doll, the greater the pressure inside the doll. This is due to the increased kinetic energy of particles at higher temperatures. The increased pressure pushes the water out through the opening and causes the doll to "pee".
Since higher temperatures will create greater pressures, the increased temperature causes the water to project further. This supports the claims made by instructors at the Chinese teahouse (in Kunming) that if the water is not hot enough to make tea, the doll would not project a long smooth water stream.
Analysis of the parabolic water path has contributed greatly to this experiment. It would be difficult to get an accurate measure of the time of flight, considering how fast the water travels.
The properties of a parabolic curve, such as the function for finding range of the curve without knowing the time of flight, allow the velocity to be calculated.
The relationship between water pressure and the initial velocity as water leaves the bottle in Part 1, can be related to the velocity of the water as it leaves the doll in part 2, leading to an approximation for the pressure level inside the doll.
Limitations and Suggestions for Further Work
There were a number of limitations in this experiment. In Part 1, the use of scotch tape to prevent the water from leaking from the hole (while making measurements) had a negative consequence. The glue from the tape seemed to remain on the bottle around the hole even after the tape was removed. This interfered with the rate that the water was projected from the bottle. More accurate data could be obtained if the weight and depth of water were predetermined, and the tape was not used.
The holes drilled in the side of the bottle presented a similar difficulty. It appeared that the holes were not perfectly round. This is due to the fact that some drilled surface remained attached to the edge of the hole. When water was pushed out of the bottle, on some occasions, the attached surface was pushed (by the water) to block the hole. A sharp object was needed to push these surfaces back in, to allow water to flow out again. These limitations could be avoided by experimenting with different drills. Sand paper could be used to remove the attached surface.
A suggestion for improving Part 1 is to drill five holes along the side of the bottle instead of one (keeping in mind that the drilled holes must be clean and completely round). Then measure the distances from the top of the bottle to the holes, which will be the depth of water. Fill the bottle up to the top and use a video camera to film the path of water as it leaves the bottle.
It would be advisable to carry out this part of the experiment with a taller container. A wider range of data would help create a more accurate approximation function for Part 2.
The major limitation in Part 2 was the fluctuation of water path and the limited data. Only three points were available. When time permits more data could be collected.
The problem of fluctuating flow was partly solved with the aid of a digital video camera (because the image of a smoothly projected water path can be selected). The fluctuations could have been due to the uneven rate of pouring the hot water on to the doll. Wind was also a possible cause of this instability, since this experiment was done outdoors. These limitations could be reduced if the experiment was done indoors, where there was not as much air movement and the rate of pouring hot water was more constant. Hot water could be supplied from a drinking tank type container at a constant rate.
Editors note: the work has been edited without altering the data, the sense or the order of treatment. The original graphs have been reproduced. There is some doubt attached to fitting linear relationships to scattered data points. It would be interesting to follow the suggestions in the evaluation and repeat the experiment to obtain more (and more accurate) data.