THERMO Spoken Here! ~ J. Pohl © TOC     NEXT    ~   116

Champagne Cork Velocity

At the Professors Emeritus Banquet, after the speeches and hand-shaking, a student bartender opened the first bottle of champagne with a loud "POP." As everyone laughed, the champagne cork flew from the bottle upward and just over a rafter in the gymnasium. The mass of the cork is one gram and the height of the rafter is about 10 meters above the champagne bottle.

Estimate the speed of the cork the instant it left the neck of the bottle. "Estimate" is the proper word; it is always the case that a precise answer can not be obtained. Approximation is required.

Scenario:  The first paragraph above describes the event.

System and Model:  Obviously, the cork is the system and it is modeled as a BODY.

Energy Equation:  The event involves changes of kinetic and potential energy of the cork. A good practice is to write the energy equation to begin, then modify it to suit the system and event being considered. We write the extrinsic energy equation as:

An increment form of the equation relates
kinetic and potential energy changes with
works that might occur.

First we inspect the "sum of works" term. We realize during flight the cork experienced friction (a drag force) as it moved through air. That friction force acted on the boundary of the cork. The friction force opposed the motion. But the cork displaced along the path of its flight: the effect constitutes friction work of the cork. This work is negative meaning a decrease of energy of the cork (as we expect). At this level of our studies we cannot calculate the effect, the frictional work of the event. Our only choice is to assume it is negligibly small ~ zero.

Also we might realize that a gravity force acts on the cork and it is displaced. Should this effect be included as work? The answer is no. The work associated with gravity was transformed into an energy term: potential energy. Work is associated with surface forces only ( learning now and explained later). Hence, ignoring the drag force of the cork moving through air approximates the sum of works for the event to equal zero.

(2) This implicit equation states that the
change of the sum of energies equals zero.

Next apply definitions of Δke and Δpe to the terms. Rewrite the energy equation as:

(3) This equation identifies the energies
in terms of system properties.

Usually the first (some say "initial") state of a system is known. For the cork, 1 is the immediate instant in time the cork goes "pop." Just exploded from the bottle, it has a speed (straight-up, we assume). We seek an approximation of this speed. The elevation of the cork at that instance, is level with the upheld bottle, which we don't know. It is some number, z1*

STATE (1):   Elevation:  zcork,1 = z*cork,1?  and    Speed: Vcork,1 = ?

We choose our event to end at the precise (imagined) time that the cork passed over the beam. Some call this the final State, or we might just label it as 2. In the event-ending condition, the cork has an elevation greater by 10 meters than initially. But its speed, at the very top of its flight becomes zero. We don't know the final elevation of the cork, so we write it as an "increase from the initial elevation."

STATE (2):    Elevation:  zcork,2 = z*cork,1 + 10 m   and  Speed:  Vcork,2 = 0

We enter these conditions into the equation to see what it will tell us.

It is best to avoid needless algebra.
Work with the equation, as is, until
only one unknown remains.

A calculation shows: V1 = 14 m/s. But we have made assumptions. We assumed no friction. Our number is a minimum value. To be precise, we realize the actual, initial speed of the cork had to be greater.

Our answer:   vcork,1 > 14.0 m/s.