### Data

As soon as I saw this product prototype I got my phone out and started dropping it! I wanted to see what the data coming from the sensors looked like. And indeed, it’s possible to identify when your phone is falling just based on the data from the accelerometers – and using the descriptive measures we’ve been studying!

Right away I collected the following short data sets with my phone:

- One when I the phone was sitting flat on my kitchen table
- One when I was standing and holding the phone as I would when looking at it normally
- One when I was lowering the phone from standing height to the ground, as if to put it in my bag
- One when I dropped my phone as the inventor does in the video above

For the questions below, you should download the following Excel file which contains those four data sets in different tabs of the spreadsheet (all measurements are in m/s^{2}). Note that if you’re using Google Sheets then you can upload this and/or open it there:

Each tab of this spreadsheet contains the accelerometer data measured in each of the three perpendicular axes of my phone over a period of a few seconds. Each column corresponds to a different axis and each row corresponds to a different instant in time (they are roughly 0.02 seconds apart).

Accelerometers measure specific force (which can be thought of as the acceleration due to gravity being experienced by the phone *plus* the acceleration due to any motion the phone might be undergoing). So what you have in that file is the total acceleration (gravity + motion) being experienced in each axis of the phone. This is measured every 0.02 seconds for around three seconds for each of the cases described in 1, 2, 3, and 4 above.

(You *don’t* need to collect your own data for this problem but you can play around with your own phone using an app like this one » if you want to.)

### Questions

a) Use Excel or Google Sheets to create a scatter plot of each set of data – i.e. create a total of four plots, with each plot containing the accelerometer data in each of the three axes. Label your plot appropriately and include a screenshot or image of your plot in the file you submit.

b) Identify which plot corresponds to which of the above data sets, i.e. which of your four plots from a) corresponds to each of the cases 1, 2, 3, and 4? You should be able to do this by thinking through the motion implied by each case I’ve described and by looking at the accelerometer data itself in the scatter plots.

c) Add a new fourth column to each of the four data sets by computing the magnitude of the specific force (total acceleration) vector at each epoch (i.e. at each moment in time). As you will have learned in applied math, the magnitude of a vector is computed as the square root of the sum of the squares of the values in each component.

d) For the data set where the phone was sitting on the table:

i) Create a scatter plot of its magnitude value – the one you calculated in c).

ii) When thinking about the measurement model, what errors are you likely seeing here if the true magnitude of gravity at the location of my table is 9.83245 m/s^{2}? (It might help to subtract this value from the data before plotting it.)

e) For the data set where I dropped my phone:

i) Create a scatter plot of its magnitude value – the one you calculated in c). What do you see?

ii) Briefly explain the physics going on here and how it shows up in the measurements being taken.

iii) Also explain what is the key characteristic or event in the data that identifies the time at which a phone is dropped?

f) Tell me which one of the following descriptive statistics would be most useful for identifying a dropped phone from the magnitude data, and why:

- Mean
- Standard deviation
- Range

Feel free to reinforce your answer to this with values computed from the magnitude data.

g) Fill in the blanks in the following sentence so that it would serve as a condition statement in an algorithm for triggering the springs that protect a phone, e.g.:

If [this descriptive statistic] calculated for [this type or column of data] [meets this condition] in any [#-second] period of time, then the user likely dropped their phone, so release the springs right away in order to protect the phone!