Math 355-01 – Pro ject – Pop-Pop Boat • This assignment constitutes 10% of the course grade and will be graded out of 10 points. • You should turn in a well-documented spreadsheet with appropriate graphs. • Any student who builds a functional pop-pop boat

Math 355-01 – Project – Pop-Pop Boat

• This assignment constitutes 10% of the course grade and will be graded out of 10 points.

• You should turn in a well-documented spreadsheet with appropriate graphs.

• Any student who builds a functional pop-pop boat will gain 2 bonus points, equivalent to 2% of the course grade.

• The student who builds the fastest pop-pop boat will earn a special prize.

Project Description

The goal of this project is to use numerical methods to implement a simple model for the mo- tion of a pop-pop boat, similar to the one depicted at right. The pop-pop boat uses a simple steam engine for locomotion, but has no moving parts.

Basic Principle of Operation

1. Heat from the candle eventually causes some of the water in the boiler to flash into a steam bubble, which takes up significantly more volume than the water it replaces.

2. The creation of the bubble forces water out of the exhaust tube(s) and the boat is pushed forward in the process.

3. The steam bubble cools rapidly once outside the boiler and after a short amount of time it will condense, creating a void in the exhaust tube(s).

4. A vacuum exists initially in the void and pulls water into the exhaust tubes while also pulling back somewhat on the boat.

5. The cycle is now complete and will restart once the heat builds up enough to produce another steam bubble.

Mathematical Model The mathematical model considered in this project ignores many aspects of the real system in favor

of a simple set of equations that captures the essential dynamics. The model focuses on two important components of the pop-pop boat: (a) the oscillation of a water column in the exhaust and (b) the propulsion of the boat due to the jet of expelled water from the exhaust. The system of differential equations are

dx

dt = v

dv

dt = h − k1x − k2v

dX

dt = V

dV

dt = k3(v+)

2 − k4V,

where x(t) represents the displacement of the water column in the exhaust tube, v(t) is the oscillation speed of the water column, X(t) is the distance travelled by the boat, and V (t) is the speed of the boat. Note that v+ is defined as follows,

v+ =

{ v v ≥ 0 0 v < 0.

The use of v+ is related to the fact that water expelled from the exhaust produces a thrust force on the boat, while there is a much smaller force pulling on the boat as water enters the exhaust tubes.

The role of the constants will now be explained.

• The constant h > 0 represents heat added to the boiler, which has a tendency to produce steam and, consequently, displace the water column.

• The constant k1 > 0 accounts for two effects. First, as steam moves farther away from the boiler, the rate of condensation will rise, forcing the water column to recede. Second, any air or steam behind the water column will be compressible (i.e., springy) and, thus, resistant to displacement.

• The constant k2 > 0 quantifies small frictional losses as the water column oscillates in the exhaust tubes.

• The thrust produced by the expelled water column is proportional to (v+)2 and the constant of proporationality is k3 > 0.

• The constant k4 > 0 represents drag forces on the hull of the boat as is slides through the water.

The following values will be assumed for the various constants for the remainder of this assignment.

h k1 k2 k3 k4 2 250 0.001 1.3 0.07

Project Objectives

1. Set up a spreadsheet to obtain an approximate solution of the differential equations using the improved Euler’s method over the time interval 0 ≤ t ≤ 20. The oscillations in the boiler occur rapidly and this will necessitate a small step-size ∆t. Determine an appropriate value for ∆t by starting with 0.1 and reducing it until the solution behaves nicely. Insert separate graphs depicting each variable as a function of time.

2. Determine the maximum speed of the pop-pop boat (v is in units of meters per second) and convert this to inches per second. What effect does increasing the heat constant by 25% have on the maximum speed? Illustrate this with your spreadsheet.

3. Determine the oscillation frequency of the boiler using the given values for the constants. Is the frequency affected by the heat constant h?