Water Spouts—Kinetic vs. Potential Energy

Introduction

Scientific discovery often begins with observations that spark curiosity. If a full water tank has three outlets—one at the top of the tank, one in the middle and one near the bottom of the tank—will the water flow from the three different outlets in the same manner? This “spouting bottle” demonstration is a visual way to have physics concepts spout in the classroom!

Concepts

• Potential vs. kinetic energy
• Bernoulli’s Law
• Torricelli’s Law
• Projectile motion

Background

A spouting bottle will be made that demonstrates Torricelli’s law. Three holes are made in the bottle at different heights. Each outlet will exhibit a different flow pattern dependent on the height. The results will be interpreted in terms of gravity and potential versus kinetic energy.

Materials

Safety Precautions

Use caution when working with hot objects. Food dye will stain clothing. Wear safety glasses. Wash hands thoroughly with soap and water before leaving the laboratory. Follow all laboratory safety guidelines.

Disposal

Please consult your current Flinn Scientific Catalog/Reference Manual for general guidelines and specific procedures, and review all federal, state and local regulations that may apply, before proceeding. Once completely cooled the paper clip may be recycled or thrown in the trash. The spouting bottle may be cleaned, dried, and stored for future use.

Prelab Preparation

Spouting Bottle

1. Find a vertical seam on the bottle or draw a vertical line on the bottle using a marker and ruler (see Figure 1).
   {12160_Preparation_Figure_1}
2. Using the vertical reference, place three dots in marker on the bottle. Using a ruler, measure the first dot 5 cm from the bottom of the bottle, the second dot 10 cm from the bottom and the third dot 15 cm from the bottom (see Figure 2).
   {12160_Preparation_Figure_2}
3. Unfold a paper clip so that one end is straightened (see Figure 2).
4. Locate a marker dot on the bottle.
5. Heat the straightened end of a paper clip by holding the end of the paperclip in the flame of a butane safety lighter for 15 seconds.
6. Use the hot end of the straightened paper clip to pierce the bottle at one of the dots, producing a small hole.
7. Repeat steps 5–6 for the other two dots (see Figure 2).
8. Using tape, attach the ruler to the bottom of the bottle. Tip: It will be easier to measure the distance the water stream travels if the ruler is placed such that a whole number on the ruler corresponds to the edge of the bottle.
9. Holding the bottle over a sink or basin, fill the bottle with water and cap it. (Optional) Add food coloring to the water.
10. Lay the bottle down on its side with the holes facing upward. Secure the bottle so that it does not roll. Note: The bottle should not leak in this position (see Figure 3).
    {12160_Preparation_Figure_3}

Procedure

1. Ask students to predict the pattern of water flow from the holes when the bottle is placed vertically. Sketch the predictions on the Water Spouts—Kinetic vs. Potential Energy Worksheet.
2. When ready for the demonstration, hold the bottle vertically over a sink or basin so the water will drain into the sink. Remove the bottle cap and observe the distinct pattern of water streaming out of the three holes (see Figure 4).
   {12160_Procedure_Figure_4}
3. Use assistants to measure the distance at which each water stream crosses the ruler and also the water level in the bottle as this happens. These numbers may vary as the water level in the bottle lowers so multiple trials may be necessary to get accurate data. (Remember: When calculating the distance, only measure the distance from the base of the bottle to the water stream (see Figure 5).
   {12160_Procedure_Figure_5}

Student Worksheet PDF

12160_Student.pdf

Teacher Tips

• This kit contains enough reusable materials to perform the demonstration as written at least seven times: three 1-L plastic soda bottles and three rulers.
• Manufacturers do occasionally change their bottle design. Holes should only be poked on the sides of the bottle perpendicular to the table or sink it sits on. Never poke holes in the curved portion of the bottle near the bottom or top. If adjustments are required, simply make sure the holes are equal distance apart.
• Ensure the paper clip is hot enough so that it makes a very small and clean hole in the plastic bottle. If the paper clip does not immediately go through the bottle, heat it for another 20 seconds. Use the same technique to make all three holes so that the holes will be the same size.
• The bottle seam provides a good vertical reference for aligning the holes, but the holes may be vertically aligned anywhere on the bottle.
• If other bottle sizes and shapes are used, space the holes equal distances apart, leaving an adequate amount of space at both the top and the bottom of the bottle. A 2-L bottle can be used for comparison to show students the relation of water flow (distance) to height.
• When measuring the vertical height of the water in the bottle, make tic marks on the bottle so that students can get a quick visual estimate of the water height. Adding water steadily from a cup or faucet may keep the water level high and fairly constant.
• As an extension, have students derive Equation 10 from the two projectile motion equations.
• Extension products available from Flinn Scientific for Bernoulli’s principle include the physical science activity Introduction to Bernoulli’s Principle, Flinn Catalog No. AP6904, and the instructions for the Bernoulli Demonstrator, Flinn Catalog No. AP5933.

Sample Data

Answers to Questions

Predict the water flow pattern by drawing in the lines on the following diagram.

Discussion Questions and Calculations

1. Sketch the observed water flow pattern on the following diagram.
2. Did your prediction (hypothesis) match the demonstration? Is a hypothesis supposed to match the data? Explain.

   My prediction did not match what happened in the demo. The hypothesis should have no bearing on the data. The hypothesis is an educated guess and the data can help support or disprove the hypothesis.

3. At which outlet does the water have the most potential energy? At which outlet does the water have the most kinetic energy?

   The water at the highest outlet on the bottle has the highest potential energy. The water at the lowest outlet on the bottle has the highest kinetic energy.

4. Calculate the initial horizontal velocity of water flow from the three different outlets using Torricelli’s Law.
   {12160_Answers_Equation_10}
5. Based on the initial horizontal velocity and the projectile motion equation (d = v_ht), explain why the water from the lowest spout traveled the longest distance.

   The water coming out of the bottom spout had the greatest initial horizontal velocity, which is also proportional to the distance the water spout travels. Therefore the lowest water jet traveled the greatest distance.

Discussion

Evangelista Torricelli (1608–1647) was a contemporary of Galileo and is probably most famous for his development of the barometer. The unit of pressure—torr—was named in his honor. Torricelli also developed a theorem or law describing the mathematical relationship between the flow rate of fluid from a tank outlet and the height of the fluid above the outlet. The liquid flowing from an outlet in a tank has the same horizontal speed as an object falling freely from the level of the liquid’s surface to the level of the outlet. This relationship was later found to be a specific case of Bernoulli’s principle (Daniel Bernoulli, 1700–1782). The usual form of Bernoulli’s equation is shown in Equation 1.

The following general assumptions are made concerning the fluid for this equation. (1) The fluid is incompressible (not easily compressed). (2) It is a nonviscous fluid—there is no friction. (3) The fluid follows a streamline pattern—there is no turbulence. (4) The fluid has a constant density. To eliminate the constant in the equation, the equation can be used by setting the condition at the top of the bottle, condition t, and the condition b, corresponding to the bottom of the bottle, equal to each other, resulting in Equation 2. [Assume condition t is the water surface on the top of the bottle and condition b is the outlet at the bottom of the bottle (see Figure 6).] At the top of the bottle the velocity is assumed to be 0. This reduces the equation to Equation 3. At the bottom of the bottle the reference height is set to equal 0 so the equation further reduces to Equation 4. Reducing the equation to simplest terms by eliminating p/ρ from both sides results in Equation 5. Equation 5 can be used as is: gh = v²/2, or it can be rearranged algebraically to solve for v, as Torricelli’s Law is commonly written, as shown by Equation 6. From Equation 6 the velocity of the water flow from each opening of the bottle can be calculated. Note that this is the same speed that an object would have when falling from a height, h, from rest.

There is a distinct pattern of water flow from the outlets in the bottle. This is because, as Galileo proposed and Newton proved, all objects fall toward the Earth at the same increasing rate (in a vacuum). That is, all objects will accelerate toward the Earth equally, regardless of their mass. In a vacuum, where there is no drag, or friction due to air, a heavy hammer will fall at exactly the same rate as a light feather. (This was demonstrated during the Apollo 15 moon landing, when David Scott dropped a hammer and feather at the same time, watched them hit the lunar surface at the same time, and then proudly announced that Galileo was correct!) At the surface of the Earth, the average acceleration toward the center of the Earth experienced by all objects is measured to be 9.8l m/s² (32 ft/s²).

The vertical distance an object will fall under the influence of gravity in a given amount of time can be calculated using Equation 7. The horizontal distance the water travels from each outlet can be estimated using the projectile motion equation, Equation 8. Solving Equation 7 for t gives √2h_o/g and substituting for t in Equation 8 gives a final estimate of the distance the water should travel given a value of h_o, Equation 9.