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Asymptotic Analysis - Vehicles Race and Algorithm Running Time

First, watch this video:

Now, let's assume the vehicles, namely the Jet, the Car, and the Motorcycle are your algorithms.
Also, suppose the length of the racecourse is the size of the data **n**.

Consequently, the speed of the vehicles can be analogized to the running time of the algorithms.

Eventually, if we had to embody the circumstances of this interesting race, we will obtain this (approximately):

Supposedly, the respective vehicles speed closed-form:

Which means:

__Formal Analysis:__

Let the instant between 0:24 and 0:30 be represented by n0, instant 0:33 represented by n'0, and instant 0:36 by n''0.
The most interesting case to study, is Tcar(n), as it grows faster than TmotorC(n) but slower than Tjet(n).

__Critique:__

We notice that, if the length of the courserace was small, the motorcycle would have won this contest.

The speed of the jet is exponentially larger than both the car and the motorcycle, but we can't realize this fact only within a sufficient distance (large data).
Indeed, the motorcycle is fast at the beginning, but it gets exceeded by the jet (**instant 0:33 of the video**).

On the other hand, the car too, when reaching its speed apex, could overstep the motorcycle (**instant 0:36 of the video**), but, it was faster than the jet until the latter could transcend the former at a certain instant between 0:24 and 0:30.
We can pretend, then, that the speed of:

- the __jet__ is **exponential**,
- the __car__ is **quadratic**,
- the __motorcycle__ is **linear**.

Also, we can assert we don't really need such a dispositive (like the one in the video) to show that a bike can be faster or slower, between a certain distance range.

But, we may image that, for instance,

__Acknowledgment:__

a big thank you to Dr. Naeem Nizar Sheikh, who spoke about such an example during a conversation, a couple of years ago.
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