Pre-Lab Week 3 Submission
Speed of Sound with Error
During lab 1C, you calculated the speed of sound from the slope of the time-of-flight vs.
distance plot, using the equation 𝑣𝑠 = 2𝑑/𝑡.
1. Using your data from lab 1C, fit the mean time-of-flight vs. distance data to a line
and extract the fit coefficients and the covariance matrix. Print the fit coefficients
and covariance matrix, screenshot these, and present them below.
2. The covariance matrix gives the covariances between each of the fit coefficients
(Covariance - Wikipedia). Given that the error on a variable is given by the
standard deviation on measurements (square root of its variance - or
self-covariance), calculate the speed of sound and the error on the speed of
sound using error propagation. An example covariance matrix is shown below:
3. The true value of the speed of sound in air at standard temperature and pressure
is about 343 m/s. We typically say that a measurement agrees with a prediction if
the measurement is within two standard deviations of the prediction. Does your
measurement agree with the prediction of 343 m/s? Why or why not?
Yes because my prediction is within the first standard deviation of the error.
Acceleration Fitting
In unit 2, we will be exploring the acceleration of carts moving along an air track. To
analyze this motion, we will have to fit the position of the air track vs. time using the
acceleration of the cart, which has a quadratic relationship.
1. We throw a ball upwards from a height of 2.0 m at a speed of 3.0 m/s. We then
measure the height of the ball from the ground as a function of time. Write a line
of code to extract the fit coefficients of this relationship using polyfit.
Results = np.polyfit(time, position, 2)
2. Write the expected values for each of the fit coefficients that you would obtain
from polyfit if you fit this data to the appropriate function.
results[0] = -4.9, results[1] = 3, results[2] = 2