# Galilei’s Tank

As shown in Figure 1 below, the so-called *Galilean trough* consists of a rectilinear channel inclined by an angle and having an approximately semicircular cross-section.

As shown in Figure 1 below, the so-called *Galilean trough* consists of a rectilinear channel inclined by an angle and having an approximately semicircular cross-section.

In EXPERIENCE LAND MATHEMATICS, the trough, i.e. the Galilean trough, is a blue tread with a length of , which is inclined at the angle with respect to the horizontal. The approximately semi-circular cross-section of the channel, which is constant over the entire length, is “*bounded upwards*” by an “*imaginary*” straight line. The distance it generates has length as shown in Figure 2 below:

There are small obstacles in the “*bottom*” of the Galilei trough so that a ball starting at a right angle (to the trough) and at the top of the right edge (in the direction of travel) of the Galilei trough does not touch it as it travels “*down*“, if a suitable start of the ball is made. For this, there are 10 different starting possibilities as “*small*” rectilinear depressions attached perpendicular to the Galilei trough (see figure 2). The experimenter now has the task of finding the launch option that allows the ball to travel downwards “*undisturbed*“, i.e. without touching the obstacles. The path traversed by the sphere then approximately represents a *“distorted” cosine-shaped* movement.

The motion of the sphere is approximately the motion of a point mass on a two-dimensional surface, represented by a Cartesian coordinate system. The (positive) -direction describes the direction of the valley bottom running from top to bottom and the (positive) -direction describes the direction running perpendicular to it from the valley bottom to the right (in running direction) upper edge of the Galilei trough (see following figure 3):

The motion of the sphere in -direction is then determined — neglecting friction — by the *displacement-time law*

for (where denotes the *time* in seconds). Where is the *acceleration due to gravity* and is the *angle of inclination* of the Galilean trough. In the -direction there is approximately a *harmonic oscillation* (“*cosine oscillation*“) of the form

before. Here denotes the *angular frequency* of this oscillation and the *amplitude* with .

Rearranging the equation according to time, we get:

Consequently, for all time points one obtains the path-time law in -direction as a function of the path-time law in -direction:

for , i.e.

and thus

with and . So with the concrete values for the angle of inclination , the amplitude and the angular frequency :

Then the trajectory of the ball rolling “*downwards*” is approximately described by the curve shown in figure 4: