## Introduction

In this project we present a qualitative analysis of the motion of a
hydroplane racing boat like the one shown in the image above. In this model we will
focus only on its forward movement and will ignore any pitching, rolling and movement in
other directions. The model is constructed using the basic force balancing
equation. In particular, the total force is equal to the *thrust *generated by
the motor minus the *drag* due to water and air resistance. This gives the
differential equation

\[mv'=T(v)-W(v)\]

where \(v\) is the velocity, \(m\) is the mass and \(T(v)\) and \(W(v)\) are functions describing the thrust and drag respectively.

We will assume that the thrust is constant i.e. \(T(v) = T.\) On the other hand, the drag should be zero at zero velocity and initially increase with increasing velocity. However, because the boat rises in the water as \(v\) increases, the amount of water per unit surface area decreases and thus the drag will decrease for larger velocities. As \(v\) is increased still further the drag will again increase due primarily to higher air resistance. In other words, the graph of the function \(W(v)\) is qualitatively cubic in shape.

In the exercises that follow you will examine the consequences of this drag assumption on the motion of the hydroplane. All velocities are given in hundreds of kilometers per hour.

**Problem 1.** The assumption that the thrust *T*
is constant gives the differential equation
\[mv'=T-W(v).\]

Show that the
critical points
of this differential equation are given by \(T = W(v).\) The
plot below shows \(v\) on the horizontal axis and \(T\) on the vertical
axis. The yellow curve is the collection of
critical points for various values of \(v\). Use
this applet to describe the equilibrium velocities for various values of *T*. In
particular, how and why does the equilibrium velocity depend on the initial velocity?

**Problem 2.** Suppose the hydroplane is initially
at rest so that \(T(0)=0.\) What happens as the thrust \(T\) is slowly increased to
a high value of \(T\)? To investigate this situation we use an additional differential
equation to describe the changing thrust. In particular, let's assume that the rate of
change of thrust is constant so we get the system of differential equations

where \(a\) is a constant (In the applet below \(a=0.07\). Use the applet to explain this phenomenon. The curve for \(W(v)\) is included, to help you understand.

Problem 3.
Now suppose that instead of slowly pushing the throttle in
as for the last problem, we just apply full thrust from the
start. Explain the shape of the the velocity curve during
the acceleration period, as depicted below. Select the value
of the constant full thrust using the slider.

**Problem 4.** Now that the hydroplane is up to
speed the driver must bring the boat to a stop after passing the finish line.
In other words, the equation is now effectively
\[mv' = -0.07t-W(v).\]
By changing
the sign of *a* in Problem 2, describe how the velocity changes as the thrust
is decreased.

With the advent of HTML5, Javascript is now ready for prime time
for mathematical applications. There are
new Javascript demos
illustrating how we might use interactive web objects to
help students learn Calculus.