Consider the gantry vending machine shown below that dispenses Coca-Cola drinks. Note that the position of the carriage that conveys the bottle from the shelf down to the pickup point on the lower right side of the machine is controlled by the position of the vertical arm (which determines the “x”-position of the carriage), and by the position of the carriage along the length of the vertical arm (the “y”-position of the carriage).  Because of this, it is logical to think of the kinematics of the bottle motion from the pickup point to dropoff point are described by Cartesian components.

An engineer designing this carriage system needs to control the horizontal position of the vertical arm and the position of the carriage along the vertical arm in a way that it is able to pick up the bottle from the shelf and deliver it to drop off point. The design of this path would like take into consideration of the acceleration of the carriage along the path; that is, the acceleration of the carriage dictates the forces acting on the bottle, and one would not want excessive forces on the bottle in order to prevent a shaking of the bottle that could lead to increased internal pressure on opening.

Using video tracking software, we can generate the path of the bottle during the delivery of the drink product. Shown below are the x- and y-components of the position of the carriage as functions of time, that is, x(t) and y(t). The time derivatives of these two functions then go into the expressions for the Cartesian components of velocity and acceleration of the carriage. The shape of the path of the carriage is also shown below. From our study of the path kinematics description, we know that the acceleration of the bottle can also be viewed through the shape of the path (radius of curvature), the speed and the rate of change of speed of the bottle.