| Raingutter Boat Racing - | Breathing Life Into Your Boat |
The equation of motion for an accelerating boat chronicles the strength of competing forces along the boat's path through the water. There is a tug-of-war between the driving force of the wind (F) against the force of pressures that try to hold back the hull (D) and the sail (R).
This is like the pressure of sin holding us back as we think about its pleasures. Both temptations of the world where our "hulls" are and temptations of the spirit (poor attitudes, rebellion, etc.) where we hoist our sail try to hinder our fellowship with God.
Mathematical models use mathematical notation. The various environmental, structural and performance factors are written using short one or two letter symbols. Each factor also has physical units that relate to physical measurements that might be measured given appropriate instruments when possible. The table below presents the notations of the factors modeled.
| Symbol | Units | Description |
|---|---|---|
| F | oz | Force of breath on rear of sail |
| s | scalar | Drag coefficient of rear of sail |
| S | in2 | Area of sail |
| B | in/s | Velocity of breath hitting sail |
| R | oz | Force of air on front of sail resisting motion |
| e | scalar | Drag coefficient of front of sail |
| D | oz | Force of water on hull resisting motion |
| q | scalar | Drag coefficient of hull below the waterline |
| m | ozs2/in | Mass of the boat |
| v0 | in/s | Initial velocity of boat |
| L | in | Length of raingutter |
| p | ozs2/in4 | Mass-density of water |
| pa | ozs2/in4 | Mass-density of air |
| g | in/s2 | Acceleration by gravity |
The acceleration of a boat can be modeled using the general model for a powered vehicle if we assume a constant breath force.
The force of friction due to the boat's hull in the water is caused by the pressure drag of the part of the hull below the water line. We write
D = qpAv2/2 where q is the pressure drag coefficient, p is the water density, A is the frontal cross-section area of the hull below the water line and v is the speed of the boat.
The force on the sail is that of pressure drag in reverse:
F = spaSB2/2 where s is the pressure drag coefficient, pa is the air density (air is being blown), S is the rearward cross-sectional area of the sail and B is the speed of the breath.
As the boat speeds up air resists the movement of the boat. The pressure drag is the same as for the sail force but the variables are renamed and the cross-section may be different.
R = epaSv2/2 where e is the pressure drag coefficient, pa is the air density, S is the front cross-sectional area of the sail (its lower edge is assumed to be very near the water line) and v is the speed of the air which is the speed of the boat.
Putting these together with the signs of slowing force terms made negative we get:
Force = Sail Force - Frictional Forces
Air Water
ma = spSB2/2 - epaSv2/2 - qpAv2/2
We can now make the proper mapping between the physical factors of this model and those of the general one in the following table.
| General Model | This Model |
|---|---|
| F | spaSB2/2 |
| M | m |
| Df | Df=0 |
| k | (qpA+epaS)/2 |
| v0 | v0 |
| L | L |
| t | t |
Using this map we can apply the general model results to this specific model without having to derive the expressions all over again. Instead, the general model expressions are rewritten using the corresponding expression noted in the map table above. Then the expressions are simplified a bit to give the specific expressions we seek for this model. If any additional expressions are needed for this model, they are derived below.
The various factors above are related to one another using physical laws and analysis. Some of these relationships have special meaning in understanding the model. These as well as kinematic factors (like time, acceleration, speed and distance) are recorded in the next table as a summary of the model.
| Description | Units | Expression | |
|---|---|---|---|
| Equations of Motion | oz | ma = spaSB2/2 - qpAv2/2 - epaSv2/2 | |
| Displacement Distance | in | Y = 2m/(qpA+epaS) | |
| Initial Velocity | in/s | v0 = v0'exp(-y/Y) from coasting model v(y) | |
| Momentum Transfer | ozs/in | P = B\[(qpA+epaS)spaS]/2 | |
| Displacement Time | s | T = m/P | |
| Terminal Velocity | in/s | vt(B,q) = B\[spaS/(qpA+epaS)] | |
| Terminal Velocity | in/s | vt(y,t) = vt = Y arccosh(exp(y/Y))/t | |
| Warm-up Time | s | t0 = T arctanh(v0/vt) | |
| Warm-up Distance | in | y0 = - Y ln(\[1-v02/vt2]) | |
| Time | s | t(y) = T arccosh(exp((y+y0)/Y)) - t0 | |
| Velocity | in/s | v(y) = vt \[1-exp(-2(y+y0)/Y)] | |
| Velocity | in/s | v(t) = vt tanh((t+t0)/T) | |
| Distance | in | y(t) = Y ln(cosh((t+t0)/T)) - y0 | |
| Acceleration | in/s2 | a(t) = Y/T2cosh2((t+t0)/T) | |
| Breath Velocity | in/s | B(q) = vt\[(qpA+epaS)/spaS] | |
| Hull Drag Coef | scalar | q(b) = (spaSb2/vt2 - epaS)/pA | |
| Hyperbolic tangent | scalar | tanh(u) = (exp(u)-exp(-u))/(exp(u)+exp(-u)) | |
| Hyperbolic arctangent | scalar | arctanh(x) = ln((1+x)/(1-x))/2 | |
| Hyperbolic cosine | scalar | cosh(u) = (exp(u)+exp(-u))/2 | |
| Hyperbolic arccosine | scalar | arccosh(x) = ln(x+\[xx-1]) |
Note: When modeling the first breath in a race, v0 = t0 = y0 = 0.
Start with terminal velocity,
vt = B\[spaS/(qpA+epaS)]
Solve for breath speed, B,
B(q) = vt\[(qpA+epaS)/spaS]
To get the coefficient, q, start over with terminal velocity,
vt = B\[spaS/(qpA+epaS)]
Solve for it,
q(B) = (spaSB2/vt2 - epaS)/pA
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| Raingutter Boat Racing - | Breathing Life Into Your Boat |
| Copyright © 1997, 2000, 2002, 2004 by Michael Lastufka, All rights reserved worldwide. | |