# How Tall is that Tree?

This might be better titled “I don’t have a life, let me tell you about it.”

On the other hand, this is a good chance to see how the process of designing an experiment and then liaising the data with theory in order to make a prediction.

Twenty six of the first forty four days in 2017 have had measurable rainfall in the Bay Area.  On Sunday, I was sitting on the balcony taking advantage of a rare sunny day and admiring my twenty inch tall Norfolk Island pine. I typically keep it indoors, but I read that brown needles signal the need for more sunlight, and tragically my tree has a few needles with brown tips. Berkeley hasn’t been getting much sunshine, so I put it outside Sunday morning at the start of a nice day: 65℉ and clear.

As I sat and admired my miniature pine, I noticed in the background a rather large tree: a majestic coast redwood.

Being a millennial, I took an “artsy” snap of it for my story and added a captioned showing the little tree declare that it would someday grow up to be the size of the redwood in the background. This could be true, since Norfolk Island pines can grow to over 60 meters, but how tall is that redwood? I needed to estimate the height, so I could validate this little tree’s claims.

The nerd inside of me took over, and in a couple minutes I had searched the apartment for a protractor, given up, downloaded a protractor app on my phone, and begun estimating the angle my arm made when I pointed it straight to the top of the tree. Next, I noticed that the tree lined up remarkably well with the end of the building, so I estimated the distance from the balcony to the tree by pacing off the length of the third floor hallway (which is a good estimate, check out the diagrams below). I then used physics, geometry, trigonometry, and statistics to get a realistic estimate of the redwood’s height.

Soon, I was designing an experiment to estimate the height of the balcony. I have a tape measure, but it isn’t long enough to reach the ground, so I decided to use my physics intuition to relate the time a pebble takes to fall to the ground after being dropped from shoulder height. I can easily measure the time with a stopwatch and then apply simple one dimensional kinematics under a constant 9.8 m s-2 acceleration due to gravity.

What if I mess up with the stop watch? What if I don’t even realize that I messed up with the stop watch? Well, this is the beauty of statistics. As long as I don’t systematically always over- or under-estimate the time it takes the pebbles to fall, the central limit theorem suggests that given enough tries, the average of a bunch of those tries should approach the true time it takes for the pebbles to fall. Better yet, the uncertainty in my measurement goes down. If I quadruple the number of tests, I double the confidence in the measurement. Note that there are always unaccounted sources of uncertainty, but part of being a good scientist is to be as careful to account for as many sources of error as possible.

Here are the results of my experiment:

 Trial Recorded Time 1 1.19 2 1.31 3 1.28 4 1.14 5 1.24 6 1.37 7 1.48 8 1.24 9 1.16 10 1.40 11 1.38 12 1.21 13 1.24 14 1.32 15 1.22 16 1.26 17 1.29 18 1.25 19 1.31 20 1.25 μ 1.28 SE 0.0188

The average of these trials is 1.28 seconds, with a standard deviation of 0.084 seconds. The standard error of the mean is the standard deviation divided by the square root of the number of trials, or 0.0188 seconds, so the time it takes a pebble to fall from my shoulder height is 1.28±0.0188 seconds. The power of statistics is evident from the fact that I can measure the time it takes a pebble to fall from the balcony to within 19 milliseconds by the experimental technique of analyzing many trial experiments. Later you’ll see that this technique is much more powerful than my ability to estimate an angle.

Now, to translate that measured time into a height, I need the kinematic equation below:

This equation is derivable from Newton’s laws of motion in the case of a constant acceleration (gravity in this case). Here, Δy is the difference in height of the ground and the balcony (what I want), v0 is the initial speed that I let of the pebble with (zero in this case since I let it go from rest), and a is the constant acceleration that allows me to use this equation (9.8 m s-2 ). The time estimated above into this equation yields 8.03 meters. This isn’t a perfectly certain value due to the uncertainty in the time that I’ve plugged in, so I must use propagation of uncertainty to see how the uncertainty in my time estimate begets uncertainty in the height.

Using this equation, the uncertainty in the estimate of the height is 0.235 meters. Thus, the height of my shoulder from this balcony is 8.03±0.235 meters. What this means is that for this experiment, my scatter in the measurements of the the time for the pebble to fall translates to an uncertainty in the distance fallen to within 24 cm of the true value. This is sufficient since the tree we are trying to estimate the height of is much larger than this, and the error in other measurements will dominate.

It’s time for a couple diagrams:

First, I measured the length of the third floor hallway (20.8 m), which I then combined with the distance between the tree’s base and the side of the building (4.1 m) in order to geometrically determine the lateral distance between the balcony and the tree. The tree aligns very well with the length of the hallway from visual inspection (see above). The narrow red triangle is a right-triangle, so I can use Pythagorean theorem to get the length of the hypotenuse, which is the lateral distance between the balcony and the tree.

Next, I incorporate these measured values, which I take as absolute with no uncertainty (I used a tape measure). I then combine these measurements with the estimate for the viewing angle of the top of the tree. This angle is highly uncertain since I used a protractor on the screen of my phone. I conservatively estimated an uncertainty of 10° in this measurement, which I then propagated through the trigonometry necessary to calculate the height of the tree:

H is the height of the tree with respect to the height of the balcony, L is the lateral distance from the balcony to the tree, and θ is the angle between the lateral distance and the direction of the top of the tree (see figure below).

Again propagating uncertainty:

Again, the height H is not the height of the tree, but the height above the balcony, so I add the two heights I have estimated, and after all of the above, I estimated the height of the nearby redwood to be 41.9±13.2 meters. Note that in order to add the combined uncertainties, I added them in quadrature, but since 13.2 is much larger than 0.235, the final result is still 13.2 with the proper number of significant digits.

In summary, this was how I spent my Sunday afternoon, and the tree outside is 41.9±13.2 meters tall, which is less than the maximum height that Norfolk Island Pines can reach, so it was not incorrect for my potted pine to hope to reach the height of this redwood outside.

I hope this demonstrates the care that must be taken whenever a number that is reported in a scientific analysis.

Aside: The images above were clipped from Google Maps and its amazing 3D feature. Google’s 3D images here are not completely to scale, I assume they rely on some projection algorithm. Despite that, this simple analysis demonstrates the vast amount of information that Google has regarding our physical world (let alone all the data they have on us). I think Google should continue to work toward making this data public.