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After the speeches and hand-shaking at the Professors Emeritus Banquet," a student bartender, opened the first bottle of champagne with a loud "POP." As everyone laughed, the cork flew upward and 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 at the instant it left the neck of the bottle.
♦ The event involves change of energy change 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.
An increment form of the equation relates
Inspecting the equation we realize the event, in general, will involve change of velocity and change of elevation so the kinetic and potential energy might not be zero.
Next we inspect the "sum of works." 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.
This implicit equation states that the|
change of the sum of energies equals zero.
Next write the energy equation for the cork explicitly:
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, z*
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.
A calculation shows: V1 = 14 m/s. But we have made many assumptions. We assumed no friction and a vertical flight. 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.