The “modern physics” course has a lab where students measure the speed of sound. The apparatus consists of an air-filled tube with a sound generator at one end and a microphone that can be set at any specified position within the tube. Using an oscilloscope, the transit time between the sound generator and microphone can be measured precisely. Knowing the position p and transit time t allows the speed of sound v to be calculated, based on the simple model:

Here are some data recorded by a student group calling themselves “CDT”.

position | transit time |

(m) | (millisec) |

0.2 | 0.6839 |

0.4 | 1.252 |

0.6 | 1.852 |

0.8 | 2.458 |

1.0 | 3.097 |

1.2 | 3.619 |

1.4 | 4.181 |

Part 1.

Enter these data into a spreadsheet in the standard case-variable format. Then fit an appropriate model. Note that the relationship p = vt between position, velocity, and time translates into a statistical model of the form p ~ t - 1 where the velocity will be the coefficient on the t term.

What are the units of the model coefficient corresponding to velocity, given the form of the data in the table above?

A
| meters per second |

B
| miles per hour |

C
| millimeters per second |

D
| meters per millisecond |

E
| millimeters per millisecond |

F
| No units. It’s a pure number. |

G
| No way to know from the information provided. |

Compare the velocity you find from your model fit to the accepted velocity of sound (at room temperature, at sea level, in dry air): 343 m/s. There should be a reasonable match. If not, check whether your data were entered properly and whether you specified your model correctly.

Part 2.

The students who recorded the data wrote down the transit time to 4 digits of precision, but recorded the position to only 1 or 2 digits, although they might simply have left off the trailing zeros that would indicate a higher precision.

Use the data to find out how precise the position measurement is. To do this, make two assumptions that are very reasonable in this case:

- The velocity model is highly accurate, that is, sound travels at a constant velocity through the tube.
- The transit time measurements are correct. This assumption reflects current technology. Time measurements can be made very precisely, even with inexpensive equipment.

Given these assumptions, you should be able to calculate the position from the transit time and velocity. If the measured position differs from this model value — as reflected by the residuals — then the measured position is imprecise. So, a reasonable way to infer the precision of the position is by the typical size of residuals.

How big is a typical residual? One appropriate way to measure this is with the standard deviation of the residuals. Give a numerical value for this.

0.001 0.006 0.010 0.017 0.084 0.128

Part 3.

The students’ lab report doesn’t indicate how they know for certain that the
sound generator is at position zero. One way to figure this out is to measure the
generator’s position from the data themselves. Denoting the actual position of the
sound generator as p_{0}, then the equation relating position and transit time
is

Fit this model to the data.

What is the estimated value of p_{0}?

-0.032 0.012 0.000 0.012 0.032

Notice that adding new terms to the model reduces the standard deviation of the residuals. What is the new value of the standard deviation of the residuals?

0.001 0.006 0.010 0.017 0.084 0.128

Compare the estimated speed of sound found from the model p ~ t to the established value: 343 m/s . Notice that the estimate is better than the one from the model p ~ t - 1 that didn’t take into account the position of the sound generator.