2nd degree function and oblique release

As we study any subject pertaining to mathematics, we ask ourselves, "Where does this apply in real life?" Well then, we will see a case of practical application of the 2nd degree function, the oblique launching of projectiles. The oblique throw is a two-dimensional movement, composed of two simultaneous one-dimensional movements, one vertical and one horizontal. During a soccer match, when the player makes a throw to a teammate, it is observed that the trajectory described by the ball is a parabola. The maximum height reached by the ball is the vertex of the parabola and the distance separating the two players is the maximum reach of the ball (or object).

Let's perform an example for better understanding.
Example 1. A weapons company will carry out tests on a new type of missile that is being manufactured. The company intends to determine the maximum height the missile reaches after launch and its maximum range. It is known that the trajectory described by the missile is a parabola represented by the function y = – x

² + 3x, where y is the height reached by the missile (in kilometers) and x is the range (also in kilometers). What values ​​will the company find?

Solution: We know that the missile's trajectory describes a parabola represented by the function y = – x² + 3x and that this parable is concave downward. Thus, the maximum height that the missile reaches will be determined by the vertex of the parabola, since the vertex is the maximum point of the function. we will have

The missile's maximum range will be the position at which it returns to the ground again (when it hits the target). Thinking about the Cartesian plane, it will be the position where the graph of the parabola intersects the x-axis. We know that to determine the points where the parabola crosses the x axis, just set y = 0 or –x² + 3x = 0. Thus, we will have:

Therefore, we can say that the maximum height that the missile will reach will be 2.25 km and the maximum range will be 3 km.

By Marcelo Rigonatto
Specialist in Statistics and Mathematical Modeling
Brazil School Team

2nd degree function - Roles - Math - Brazil School

Source: Brazil School - https://brasilescola.uol.com.br/matematica/funcao-2-o-grau-lancamento-obliquo.htm