Bringing Google and R in a Simple Physics Experiment

There have been multiple theories of gravity over the centuries. From Aristotle, a Greek Philosopher to Newton’s Law that states that:

In this blog, I do not want to present another theory, but show how developments in technology such as data analysis with tools like R and Google apps can be used in an interesting manner in an experiment to estimate the value of gravitational acceleration. My target is to excite the readers to explore how multiple technologies available today can be brought together to create a new experience.

The Second Law of Motion:

The experiment involves the basic idea of free fall that when an object (in this case a cricket ball) is dropped, it falls with constant acceleration due to the gravitational field of earth and the aim of the experiment is to calculate the value of the acceleration of the object when falling from a specific height. The usage of the second equation of motion => s=ut+(1/2)at² has been tremendously vital in calculating the values of acceleration of free fall.

In this equation of motion:
1. s:= distance covered by the object
2. u:= initial velocity of the object
3. t:= time taken by the object in motion to reach the ground
4. a:= acceleration of the object

How I planned the experiment?

In the experiment, I kept in mind that just dropping the ball from a specific height would instantaneously make the value of the initial velocity ‘u’ of the object ‘zero’. Hence the above equation simplifies to s= (1/2)at². Acceleration of the body is obviously the gravitational acceleration of the object in free fall, therefore I replaced ‘a’ with ‘g’ in the equation where ‘g’ denotes the value of gravitational acceleration. Also, the distance covered by the object is the height at which I dropped the ball hence I replaced ‘s’ with ‘h’ which denotes the height at which I dropped the ball. This helped me to get the equation h= (1/2)gt². Since I am interested in ‘g’, I will now rewrite the equation as g= 2h/t².
The final equation which finds out the value of ‘g’ now contains only two variables. The height, h, from where the ball is dropped and the time, t, that is taken by the ball to touch the ground. Therefore, all I need to do is to drop the ball from a measured height, h, and record the time, t. Then, by using my final equation, I would be able to calculate the value of ‘g’, which should technically be equal to 9.8m/s². However, taking into account all the errors of measurement of h and t (which I will explain later), the value of ‘g’ will have errors.
Now, I faced the problem of measuring the various heights from where I wanted to drop the ball for the experiment. I checked for an inch tape at my house and the maximum value that could be measured from the inch tape was only 1.5 meters. This wasn’t adequate for my experiment because if I only work with this inch tape, I would only be capable of taking values of ‘h’ which would be less than 1.5 meters while there were plenty of places for me to drop the ball from a height more than 1.5 meters inside and outside of my house. For heights greater than 1.5 meters, I needed a way to measure the exact height from the ground to where I am going to drop the ball from. That’s where the Google Measure app came into play. Google Measure, an extremely useful app on the IOS and Android, allowed me to measure a range of heights — from 0.5 meters to almost 5 meters. Note, one component of the error introduced in the experiments is the measurements of height using the Google Measure app.

Taking measurements for the time, t, were done using the stopwatch feature in the Clock app on my android phone. I started the stopwatch at the same time I dropped the ball from a height and then stopped the stopwatch as soon as the ball touched the ground. Though taking the above measurements is a simple process, any person can see that it is fraught with errors — for a human being it is simply impossible to synchronize the ‘starting’ and ‘stopping’ of the stopwatch, respectively, with the ‘dropping’ and ‘ground touching’ events of the ball. Hence this is the second component of error in my experiments.

Error significance expected to decrease with height!

The two components of errors discussed above — namely, the ‘synchronization error’ and the ‘Google Measure error’ — are expected to have a constant contribution to the measurement of ‘t’ and ‘h’, respectively. As a consequence, their contribution becomes less significant with increasing height and time and that is what our experiment shows. See the graph of ‘g’ vs ‘h’ below plotted using R:

The graph depicts that when the height from which the ball is dropped is increased, the acceleration of the ball calculated decreases, approaching closer to the value of 9.8m/s². Although there are two exceptions at heights less than 1 meter, the values of ‘g’ are greater than 9.8m/s² including some points where the value of ‘g’ is really far from its true value and spaced out. This is primarily due to the ‘synchronization’ error as starting and stopping the stopwatch at really low heights becomes difficult which increases inaccuracy resulting in higher values of ‘g’. When height increases to 2 meters, we can actually notice that most of the points are more precise and accurate as they are close to each other and to 9.8m/s². Again, as height is increased beyond 2 meters, the usage of the stopwatch becomes easier producing better results of ‘t’ and ‘ g ‘ simultaneously. Heights greater than 3 meters are obviously the most accurate and precise in this experiment.

Closing Comments

We have seen how a simple experiment, which estimates the value of gravitational acceleration, can be made interesting by the use of today’s technologies such as Google Measure and R. I will present other similar thoughts in my subsequent blogs that make use of multiple new areas to solve a problem. I would like to thank my father, Manish Gupta, for helping me complete this experiment because he is the brain behind this experiment and the one who powers my brain.

Originally published at https://whenpranavblogs.wordpress.com on July 27, 2019.