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A Shot Under Pressure

super soaker water gun

Activity courtesy of TeachEngineering.org, a teacher-tested, searchable online library of standards-based lessons and activities developed by the Integrated Teaching and Learning Program at the University of Colorado’s College of Engineering. See companion activity on Bernoulli’s Principle.

Note: Teachers should err on the side of caution and make sure administrators know about and approve of letting students bring their water guns to school.

Summary

Working in groups of three, high school students use their understanding of projectile physics and fluid dynamics to calculate the water pressure in squirt guns by measuring the range of the water jets. They create graphs to analyze how the predicted pressure relates to the number of times they pump the water gun before shooting.

Grade level: 9-12

Time: 120 minutes

Cost: $2 per group

Learning Objectives

After this activity, students should be able to:
  • Use projectile motion physics to determine the initial velocity of a projectile launched horizontally.
  • Use Bernoulli’s equation to find the pressure of a fluid.
  • Collect, record, and analyze data to determine relationships among variables.

Learning Standards

Common Core State Math Standards

  • Create equations that describe numbers or relationships. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
  • Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
  • Represent and solve equations and inequalities graphically. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
  • Solve equations and inequalities in one variable.

Next Generation Science Standards

Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known. [9 – 12]

International Technology and Engineering Educators Association

Energy cannot be created nor destroyed; however, it can be converted from one form to another. [9-12]

Engineering Connection

water slide

When engineers design water amusement park rides, water fountains, or anything that involves shooting water, they use a similar modeling process to determine where water jets will land. This ensures the safety and best experience for its users. More generally, Bernoulli’s equation is used by any engineer working with fluids. For example, biomedical engineers use this principle to model the flow and pressure of bodily fluids, and civil engineers use it to determine the size of pipes needed to get water to people’s homes and workplaces, or to run industrial processes.

Materials List

For each group:
  • Water gun with a pumping mechanism (for example, Super Soaker Water Blasters; available at stores such as Target, Kmart, WalMart or online; ~$10 each); ask students to bring in their own, if possible
  • Moveable desk or table, on which to mount the water gun
  • Duct tape
  • Access to a tape measure long enough to measure 35+ feet (11+ meters), the shot distances
  • (optional) Chalk or tape, to mark off every 10 meters of the shooting range
  • Pen or pencil
  • Take Your Best Shot Worksheet, one per person
  • Evaluation & Enhancement Worksheet, one per person
  • (optional) Computers and Excel spreadsheet software for data manipulation and graphing

Introduction/Motivation

Nothing is better on a hot summer day than the thrill of a fast, steep ride at a water park. To make that ride both fun and safe, engineers must predict how fast people will move, what the forces on their bodies will be, and where they will land. Fortunately for you, projectile motion physics and fluid mechanics provide the equations they need to determine these variables. Today we’ll see how.

Procedure

Bernoulli Equation: The Bernoulli equation is an important expression that relates pressure, height and velocity of a fluid at one point along its flow. According to the Venturi Effect, a fluid’s pressure decreases as its velocity increases. The Bernoulli equation puts this relationship into mathematical terms and includes a term for fluid height. Think of water moving down a water slide. At the top (where you load), the water is slow moving, pushed only by the water behind it. When the slide drops, the water rushes down quickly, increasing speed as it falls. Thus, the velocity is also affected by gravity through height. When all these terms are related and scaled for density and gravity, we have the Bernoulli equation,

Equation: ½ pv2 + pgz + P = constant

Equation 1
where v is fluid velocity, ρ is fluid density, z is relative height, and P is pressure. Notice the constant has units of pressure as well.
Derived from energy conservation, the Bernoulli equation tells us that the sum of the kinetic and potential energy at any point along a streamline must remain constant. Technically, the equation only applies to the case of flow that is steady (not changing), has constant density, and is inviscid. Many fluids (such as the water in a squirt gun) approximate these conditions, making Bernoulli’s equation a useful model.
Finding the Pressure in a Water Gun: Students are asked to use the equations of projectile motion, combined with Bernoulli’s equation to find the water pressure. This section explains how to arrive at the solution, if you’d like to give students more guidance. Bernoulli’s equation can determine the pressure inside the water gun, but first you have to know how fast the water coming out of it is moving. Kinematics physics equations can help us find the velocity of the water.
As the water leaves the water gun, it is travelling entirely in the horizontal direction, so its total velocity vector can be reduced to horizontal velocity, or Vx . A simple relationship relates the horizontal velocity (V x) to the distance (x) it travels in a certain amount of time (t),
Equation: x = vx * t

Equation 2

The distance can be measured, but unfortunately, we do not know time. However, we can determine the time using another kinematic equation that relates time to the initial and final vertical positions of the water (y0 and yf ), the initial vertical velocity (vy0), and the vertical acceleration, ay ,
Equation: yf = yo + vyot + ½ ayt2

This equation simplifies greatly under the initial conditions set by this problem. The initial vertical position of the water (y0 ) is the table, and the final position (yf ) is the floor, so yf– y0 is simply the height of the table, or h. As we already said, the initial velocity of the water is directed horizontally, making the initial vertical velocity, vy0 , equal to zero as well. Finally, the vertical acceleration is g, or -9.8 m/s2, the constant acceleration due to gravity. The equation simplifies to
Equation: h = -1/2 gt2

Equation 4

Using algebra, we can solve for time, and then plug the result into Equation 2 to solve for velocity.
Once velocity is found, we can solve for pressure using Bernoulli’s equation. Because the Bernoulli equation equals the same constant at all points along a streamline, we can set the Bernoulli equation at two points equal to each other and use information on the system at one point to solve for information at another, as shown in the equation below:
Equation: 1/2pv12 + pgh1 + P1 = ½ pv22 + pgh2 + P2

Equation 5

Side-view sketch of a water gun, showing a horizontal blue line through the middle, representing the water flow line. Arrows point at the beginning and at the end of the blue line, at which point it leaves the nozzle before dropping down.
To solve for the pressure, two useful points to choose are the inside of the water gun (position #1) and the outside of the gun at the nozzle (position #2) because we know some of the quantities at these locations.

Copyright © 2009 ITL Program, College of Engineering, University of Colorado at Boulder.
For this particular problem, two useful points to choose are the inside of the gun (position #1) and the outside of the gun at the nozzle (position #2). These points are chosen because we know some of the quantities at these locations. We know that the velocity of the water inside the gun (v1 ) is initially zero, and that the pressure outside the gun (P2 ) is the atmospheric pressure. The density (ρ) is the density of water, and the relative heights (h1 and h2 ) are equal because we are keeping the gun level. After some algebraic manipulation, the pressure inside the water gun is easily solvable.

Before the Activity

  • Gather materials. If possible, ask students to bring their own super soaker water guns from home.
  • Make copies of the Take Your Best Shot Worksheet and the Evaluation & Enhancement Worksheet.
  • Find an appropriate location to shoot the squirt guns. It is best to secure a relatively open space with access to water and a dry concrete surface (to better see where water lands).
  • Set up a table or desk so it is level, and mark off with chalk or tape the distance from the table every meter for ~10 meters.
Photo shows a red gun-shaped toy laying flat on a table with its nozzle pointed off the table edge.
Copyright © 2009 James Prager, ITL Program, College of Engineering, University of Colorado at Boulder.
Place the water gun on the table, securing it with duct tape if unstable, making sure the nozzle is horizontal and level.

With the Students

  1. Conduct a pre-activity assessment discussion with the students, as described in the Assessment section.
  2. Have students read the background and procedural instructions on the Take Your Best Shot Worksheet, and set up their experiments.
  3. When ready, run the experiments as described in the worksheet.
  4. Have the students perform the calculations and graphing on the Take Your Best Shot Worksheet. (Optional: Have them use a digital spreadsheet such as Microsoft Excel for data manipulation and graphing.)
  5. Have students complete the Evaluation & Enhancement Worksheet.

Safety Issues

  • Do not aim squirt guns at eyes.

Troubleshooting Tips

  • Depending on your exact water gun design, it is probably best to place the water gun sideways on a flat table to ensure a horizontal shot and easy trigger access.
  • One or two pumps might not yield any results depending on the type of water gun used. If this is the case, have students start their data collection at the first number of pumps that yield results and perform trials at up to ten pumps beyond that.
  • It is important to leave some air in the water tank to make sure the water gun works properly.
  • If holding the trigger until the water stops shooting does not work, the depressurization of the chamber between trials can be achieved by opening the water tank, thereby depressurizing the system.

Assessment

Pre-Activity Assessment

Discussion: Solicit, integrate and summarize student responses.
  • Most popular water guns today use a hand pump to build up air pressure and shoot water out of a tube, often in order to soak your friends and family while looking totally awesome. How much pressure do you think it takes to have this fun? How would one even go about calculating that? The following activity answers those questions, but first, what do you think?

Activity Embedded Assessment

Activity Worksheet: Have students individually complete the Take Your Best Shot Worksheet. Review their answers to gauge their mastery of the subject.

Post-Activity Assessment

Evaluation & Enhancement Worksheet: Have students individually complete the Evaluation & Enhancement Worksheet in which they reflect upon potential sources for error in the experiment, their impact on the results, their graph shapes, and how the data could help them in real-world problem solving. Review their answers to gauge their mastery of the subject.

Activity Scaling

  • For lower grades, have students calculate and graph the pressure generated by five pumps instead of 10. Also, depending on students’ math ability, provide them with more help in doing the calculations. For instance, provide the equations in order of use and rearrange each one so that the unknowns are on the left sides, requiring minimal algebraic manipulation.
  • For higher grades, have students work out the problem solution more independently and require more analysis of the results.

Additional resources

  • African-American Inventors. Several eGFI teachers features, including Garrett Morgan, inventor of the traffic signal,
  • Bernoulli’s Principle. NASA lessons, hands-on activities, and history of 18th-century scientist Daniel Bernoulli for student in grades 5 to 8. [PDF]
  • Bernoulli’s Principle Companion TeachEngineering. org activity for high school students.
  • History of the Super Soaker. Brief history and timeline of the popular water gun’s development and its inventor, mechanical engineer Lonnie Johnson, Jr.
  • Who Made That Super Soaker? The 2013 New York Times Magazine feature includes a q & a with a University of Texas fluid dynamics Ph.D. student who designs high-power squirt guns.
  • Super Soaker Project-Based Math Activity This standards-based high-school lesson from Worcester Polytechnic Institute’s Center for Industrial Mathematics and Statistics lets students apply algebra, gather data, develop and interpret graphs, plot linear equations, and calculate percents. Click HERE for other math projects.
References

Bernoulli’s principle. Wikipedia, The Free Encyclopedia. Accessed February 21, 2016.

Knight, Randall. Physics for Scientists and Engineers: A Strategic Approach. Second edition. San Francisco, CA: Pearson Addison-Wesley, 2008.

Munson, B. R., Young, D.F., Okiishi, T.H., Fundamentals of Fluid Mechanics. Fifth edition. New York, NY: John Wiley & Sons, Inc., 2006.

Contributed by by James Prager, Karen King, Denise W. Carlson © 2009 by Regents of the University of Colorado.

Supporting Program: Integrated Teaching and Learning Program, College of Engineering, University of Colorado Boulder

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