“I jumped from 10 meters!” Or not?
Our trip to Greece was full of surprises, discoveries, fatigue, laughter and adrenaline. We visited Athens by going to stick our noses everywhere, even in the Anafiotika district. We have turned in long and wide the island of Naxos, until to find of it its more lost and fascinating little church, that one of Agios Sozon. We traveled Milos from north to south (and vice versa) to visit every beach of this island of 1000 colors. And just in Milos we jumped from the cliff of Sarakiniko risking the bone of the neck and not only that.
Here, one of the recurring memories of the trip to Greece is just that jump from the cliff of Sarakiniko. Every time we ask ourselves two questions:
1- How long have we been going up?
2- And above all: why did we do it?
While for the second question, the only answer is “ok, it went well, we don’t think about it anymore”, for the first question we are stuck with doubts from time to time. At the time of the dive we had estimated a height of 4 meters, more or less. But soon the first doubts about the “nasometric” estimate were revealed. So what to do? Nothing better than asking Galileo Galilei for help.
Brief history of uniformly accelerated straight motion
Don’t make that face, I’ll make it very brief. In 1638 Galileo Galilei published Discorsi e dimostrazioni matematiche intorno a due nuove scienze attinenti alla meccanica e i movimenti locali in which he investigated with a method – not by chance – scientifically the problem of a body falling from a given height to the surface, drawing essentially three conclusions:
A falling body moves with uniformly accelerated rectilinear motion
In the absence of air all bodies fall with the same acceleration, called acceleration of gravity, indicated by the symbol g, and which on the Earth is worth on average 9.89 m/s².
In the presence of air, due to friction, the bodies may fall with different hourly laws from those of the uniformly accelerated motion
Galileo therefore demonstrates that a body in free fall and without friction moves with a rectilinear motion uniformly accelerated, regardless of its mass.
There are two equations that describe this motion:
Well, we have what we need (I mean the second equation). Now: let’s calculate the height of this blessed dive!
Calculating the height of the dive
Having said that, let’s move on to calculation. Since we have a video of the dive taken with the iPhone, the first thing that the good scientist-traveller does is to calculate the time between the moment of departure from the cliff until the impact with the water.
Using video editing programs such as Virtualdub you can calculate (with good approximation) this time. The verdict was: 1.3 seconds. I assure you that, when you are in the air, they seem much more (for the squeeze). Something like that:
Well, at this point:
- knowing that t equals 1.3 seconds
- assuming a rectilinear motion (yes, I know that it is not exactly so, but have mercy)
- neglecting the friction with the air
- and applying the hourly law of uniformly accelerated rectilinear motion, it is obtained:
x(t) = 1/2 * g * t² = (9,89 * 1,3²)/2 = (9,89 * 1,69)/2 = 8,35 meters
The mystery is solved! Now you know how to calculate the height of your next dive. Happy, huh? Um… Hey! But… have you fallen asleep?