A plot of the average velocity vs. net force for the terminal velocity experiment is shown below.

• Average velocities are obtained by dividing the distance through which the marble moves by the time observed.
• The net force is the force in excess of that required to just barely move the marble through the fluid.
• From the shape of the curve, and from theoretical considerations, we expect that v = a * Fnet ^ p.
• We could use curve-fitting techniques to determine the most appropriate value of p.

Below we reverse the variables to show Fnet vs. vAve.

• We think of this curve as indicating the net force exerted on the marble as a function of its velocity.
• Assuming that the marble reaches a terminal velocity shortly after is released from rest, we interpret this as an approximate graph of net force vs. velocity.

Since the amount of water that must be disturbed in a given time interval is proportional to the velocity of the marble, and since the disturbed water will gain more velocity for a faster marble than a slower, we expect that the force required to move the marble through the water will be at least proportional to the square, and perhaps to the cube of the velocity.

• Using DERIVE we find that F = k v^2 tends to give us a good model for the data for each of the three groups performing the experiment.
• However, for different groups using marbles of approximately the same size we obtained different values of k.
• Our results showed that k was probably between .02 N s^2 / m^2 and .04 N s^2 / m^2 for a marble of diameter 2 cm.
• These results can be compared with results given in texts or tables in handbooks.

`00

We observe a soft drink bottle with an initial state where 300 ml of air are confined at a temperature of 298 K and a pressure of 1atmosphere.

• A thin tube leads from a water reservoir in the bottom of the bottle to a container at some height above the reservoir.
• When thermal energy is added to the gas in the system, the pressure in the system will increase.
• As the pressure in the system increases, water will be pushed up the tube.
• A sufficient pressure increase will transfer water to the upper container, doing significant work as a result of the added thermal energy.
• The rise of water in the tube can be seen as the direct result of Bernoulli's equation, with the `rho g h increase of the water in the tube balances the pressure increase at the water surface in the bottle.
• The relationiship between the added thermal energy and the work accomplished is the beginning of our study of thermodynamics.

To raise the water to height `dh above the water surface in the bottle requires that the gas achieve a sufficient temperature to supply the required pressure.

• The water at the top of the tube is at atmospheric pressure.
• The difference in `rho g h between water surface and the top of the tube is `rho g `dh.
• The pressure at the water surface must therefore be `rho g `dh greater than at the top of the tube. 
• So the pressure at the water surface is 1 atm + `rho g `dh.

`01

Since the system, originally at 298 K, was immersed in water at a temperature of 70 Celsius = 343 K, we assume that the temperature of the gas rose to near 343 K.

• The pressure therefore should have increased by the same factor as the temperature, a factor of 1.15.
• The pressure in the heated system should therefore have been 1.15 atm.

We estimated the height which the column was raised to be approximately 2 m.

• The difference in `rho g h corresponding to a 2 meter water column is easily found to be 20 kPa, or about .2 atm.
• From this estimate of the column height we see that the pressure in the heated system should have been about 1.2 atm.