This post focuses on learning numerical methods from movie clip examples. Specifically, the nonlinear equation describing Johnny Utah’s jump in the movie Point Break is worked out.
In the opening sequence of the movie Point Break (Edgar Ramirez, 2015), the protagonist, Johnny Utah, an extreme sports athlete, makes a high-powered motorcycle descent down a rocky mountain – reminiscent of scenes from James Bond movies. The descent concludes where the hill ends, but a few meters ahead, another mountain, a little one resembling Tepuyito) is present but separated by a precipice. As expected, Johnny manages to jump from one cliff to another, as seen in the scene starting at minute 1:23.
A similar problem that allows predicting the conditions under which Johnny achieves the jump is related to the trajectory of a ball thrown from one point to another. This trajectory is defined by coordinates (x, y), as illustrated in the figure below.
The trajectory can be modeled using the projectile motion equation:
$$ y=\tan(\theta_o)x-\frac{g}{2v_o^2{cos^2(\theta_o)}}x^2 $$
The above equation is nonlinear in terms of θo, and numerical methods are required to solve it. The Newton-Raphson method is a numerical method that can be applied to solve Equation 1. To apply the above equation to the Johnny Utah’s jumps, we assume that the level from which the jump starts is equal to the level of the second mountain (y = 0), as shown below.
With the projectile motion equation, Johnny can know at what angle (θo) he should position the motorcycle if he wants to save his life. To apply the equation, he only needs to read the speedometer, the speed at which the motorcycle is traveling before the jump (vo), and know in advance the distance between the mountains (x).
The Newton-Raphson method requires an initial guess (providing a tentative solution value), and iteratively, the method finds the solution of the equation, in other words, the value that saves Johnny. What initial estimate would you take in this problem? Are you up for solving it?
We look forward to your comments. In a subsequent post, we will present the code that solves the problem and its implementation in code language R.