With gas prices as high as they are and having a growing concern for the environment, Americans today are becoming conscious about things they can do to improve fuel efficiency. Many realize that driving at the posted speed limits promotes both safety and reduces the rate at which fuel is consumed. With these things in mind, some have purchased hybrid vehicles including the Toyota Prius, all-electric vehicles such as the Nissan LEAF, or range-extending vehicles such as the Chevy Volt. Those with EV's soon realize that they get more miles per charge if they avoid driving at excessively high speeds on the open road.

With this background as a base, it would be very instructive if students could perform a laboratory investigation that would provide a quantitative relationship between distance per unit of fuel consumed (a measure of efficiency) and speed. One way to accomplish this is by investigating these factors using an N-scale electric train set, such as those sold by Bachman for around $100, and commonly available in toy and hobby stores. In this case, fuel is the unit of electrical energy, i.e. the kilowatt-hour (kWh), or on the scale of a hobby train, more appropriately the watt-second (W-s). The picture below shows a train set, with a power supply providing the electrical power to run the train, The power supply provides readings for voltage (V) in volts and current (I) in amps, from which power can be calculated by the product VI. A meter stick is used to measure the diameter of the circular train track, from which the radius is found to be 0.285 m. PocketLab is mounted to the top of the engine using tabs that come with the PocketLab Maker Kit.

Starting with a low voltage, the train was allowed to run for about one lap. This process was repeated for a sequence of voltages to slightly above the train manufacture's recommended limit of 16 volts. Voltages can be read from the video. PocketLab was set to provide angular acceleration. With PocketLab mounted on the engine with the train moving in the XY plane, Z angular acceleration (shown in green in the video) is the variable of interest, telling us the number of degrees per second that the train revolves on its circular track. It is important to make sure that angular acceleration is zeroed prior to capturing data and video with the PocketLab app.

The image shown below contains a graph of Z angular velocity vs. time from data produced by the PocketLab app. The graph was obtained by importing data from the PocketLab app into Logger Pro, an exceptional educational scientific analysis software from Vernier Software & Technology (vernier.com). The statistical analysis capability of Logger Pro is used to obtain mean angular velocity for each of the voltage steps during the experiment. This is an extremely useful capability as the angular acceleration does vary a fair amount at each step level. For example, the graph shows that the mean Z angular velocity for the second voltage step was 32.59 degrees/s. The step nature of the Logger Pro chart is a result of the voltage being increased to a new level after a few laps have been made by the train at a given voltage.

The Microsoft Excel table below summarizes all of the raw data as well as come calculated columns containing power, period, speed, energy per lap, and efficiency. A copy of the Microsoft Excel file is attached so that you can look at the formulas used in these calcualted columns. Power is voltage times current (P= VI). Period is 360/angular velocity. Speed is 2*Pi*R / period. The energy per lap in W-s is power multiplied by period. Efficiency in Laps per W-s is the reciprocal of Energy per lap. Therefore, efficiency for our N-scale train is measured in laps per W-s. This is analogous to mpg for a traditional gas consuming car, or to miles per charge or miles per kWh for an EV.

The final task is to construct a graph of efficiency vs. speed. The Excel graph is shown in the figure below. The graph appears to have a parabolic shape suggesting that a quadratic might be appropriate. The efficiency starts off very low at low speeds, reaches a peak around 0.4 m/s, and then begins to drop at higher speeds. The value 0.9938 for the correlation coefficient indicates a very good fit of the quadratic to the data.

Questions for Students:

1. Why is the efficiency of the train so low at low speeds?

2. What are some possible explanations for the train's efficiency dropping after reaching the speed of peak efficiency?

3. Considering a moving automobile as an analogy, (1) What is the efficiency of a car when it is at rest in a traffic jam and why? (2) What factors reduce the efficiency of a typical car at very high speeds?