View Video Clip

Note all information given on the video clip.

• Determine how much thermal energy was lost by the water originally in the cup as it cooled to 0.1 Celsius (recall that for a gram of water to change by 1 Celsius the water must either gain or lose 4.19 Joules of thermal energy).
• Determine how much thermal energy must therefore have been gained by the snow, assuming that no energy was exchanged except between the snow and the water.
• Determine how much thermal energy was gained by each gram of snow in the melting process.

View Video Clip

Obtain temperature vs. clock time data for the two temperature probes, taking readings about every 30 seconds.

• Sketch a good graph of the temperature vs. clock time curves, with both curves on the same set of coordinate axes.

Determine halflife of the first probe:

• Determine the time required for the smaller (faster) probe to approach halfway from its initial temperature to the 75 degree Fahrenheit room temperature.
• Determine the time required to then approach halfway to the room temperature from this temperature.
• Determine the time required for the next halfway approach.
• Determine the time required to approach halfway to room temperature starting at 20 degrees Fahrenheit.
• Average the halfway times to estimate the halflife of this probe.

Determine the time required for the larger (slower) probe to approach 25% of the way from its initial temperature to room temperature.

• Determine the time required to approach 25% of the way to room temperature from that point.
• Determine also the time required to approach halfway to room temperature from the initial temperature.
• Why is this time not equal to the sum of the two 25%-approach times?

Create a mathematical model of the behavior of each probe:

• The time required to approach halfway to room temperature is called the ‘halflife’ of the temperature. Theoretically, according to Newton’s Second Law of Cooling, the halflife for a probe should be the same regardless of the starting time or temperature.
• A mathematical model of temperature is then T = (T0 – Tr) 2^-(k t) + Tr, where T0 is the initial temperature, Tr the room temperature, and k = 1 / halflife. An alternative model is then T = (T0 – Tr) e^-(k t) + Tr, where now k = ln(2) / halflife  = .693  / halflife, approximately.  Either model works fine; the models are in fact identical.

Evaluate your models by comparing with your observations:

• Using either model of the temperature, determine the temperature of the smaller (faster) probe as predicted by the function for each of your observed clock times. Compare with observed temperatures and report how close the model is to the observations.
• Do the same for the larger (slower) probe.

Answer questions about your observations:

• At what temperature are the temperatures of the two probes most different?
• At what temperature do the two probes appear to be changing at the same rate?

Answer similar questions about your models:

• Using a graphing calculator or a computer sketch on one set of coordinate axes the graphs of the functions representing the two probes.
• Find the temperature at which the model predicts the greatest temperature difference.
• Find the temperature at which the two temperatures are changing at the same rate.
• (University Physics students: use calculus to determine the exact temperature at which the rates of change are identical).

View Video Clip

Collect data, sketch a depth vs. clock time graph and estimate rates of depth change for flow from the 1/8-inch hole:

• For flow from the 1/8-inch hole obtain depth vs. clock time data (clock time starts when the water surface first reaches the 30 cm mark and keeps running—we aren’t graphing time intervals) for the water flowing from the cylinder.
• Sketch a graph of depth vs. clock time and estimate the rate at which depth is changing at the instant the depth is 30 cm, at the instant depth is 20 cm, at the instant the depth is 10 cm and at the instant the depth is 5 cm.
• Using the rates of depth change determine the velocity of the water exiting the hole.

Repeat for water flow from both holes:

• Take similar data for water flowing simultaneously from both the 1/8-inch and 3/16-inch holes, sketch a graph analogous to the previous and estimate rates of depth change at the same points as before.

Determine water velocities using cylinder diameter (1.5 inches) and hole diameters:

• Using the rates of depth change determine the velocity of the water exiting the holes, assuming that water exits both holes at the same velocity (this is a first approximation which is pretty accurate but not completely so—there is more frictional effect with a smaller hole, since on the average the exiting water is closer to the edges of the hole where most of the friction originates).

For each of the two systems measured (i.e., the 1-hole and 2-hole systems), construct a mathematical model and infer the velocity of the exiting water:

• Using a spreadsheet, a graphing calculator or a computer algebra system determine the quadratic function which best models the four (x=clock time, y=depth) data points corresponding to depths 30, 20, 10 and 5 cm. (see next paragraph for instructions on how to do this using Excel).
• Determine the rate at each of these clock times:

• If the quadratic function is y = a t^2 + b t + c, then at clock time t the rate of depth change is 2 a t + b. For the rate at each clock time determine the velocity of the exiting water, assuming that at any given instant all the exiting water travels at the same velocity.

• To use Excel to determine the quadratic function, place your data in two columns to represent depth vs. clock time. Using the Chart Wizard to create a scatter graph, points only, of your data. Right-click on a data point and choose Add Trendline. Choose Polynomial, order 2. Under Options choose Display Equation on Chart. Click OK and the quadratic model will appear on your graph.

View the video clip of the water stream and the ball. 

• The ball descends from an altitude equal to the water depth, so its PE change per mass unit is the same as the PE change in the water per unit of exiting mass, and we would therefore expect the velocities and hence the projectile paths to be identical. 
• There are second-order effects, mostly from friction and the rotational KE of the ball, which for this situation have very nearly identical effects on the observed final velocities of the respective systems.

View Video Clip

From the video clip note the diameter of the sphere, the masses of the weights to be added, the total distance traveled by the sphere and the distance between the measuring posts. Then determine the time required for the sphere to travel the entire distance, and the time to travel a distance corresponding to the distance between the posts.

• The release of the sphere is indicated by a sudden vibration of the thread, clearly visible in the video clip. The time for the total distance is measured from this start to the end of the last post, and can be measured by a synchronized pendulum, a stopwatch or the TIMER program.
• The final velocity is measured by counting frames in the frame-by-frame picture as the marker on the thread passes between the two posts.

Analyze the motion of the sphere:

• Determine the average velocity and average acceleration of the sphere.
• Determine also the acceleration the sphere would have had under the influence of the net force initially exerted on the system consisting of the sphere (which is full of water) and the counterweight (the system initially behaves like at Atwood machine; as it gains velocity the fluid drag exerts its force).

Judge whether terminal velocity was attained in each situation:

• Sketch a plausible velocity vs. clock time curve for the motion of the ball, with the initial slope equal to the initial acceleration determined above, and with final velocity equal to the measured value.
• From your curve determine whether it is plausible that the final velocity is indeed the terminal velocity of the system, or whether a significant velocity increase should be expected.