Exercise 3: Projectile During the "classico" which is a football match between two great teams Real Madrid and Barcelona, and at the minute 92 when the result between the two teams was 2-2, Messi has seen that the goalkeeper of real Madrid Navas is in an advanced position with respect to its goal, then Messi decided to score a goal by sending the ball above Navas located in a point A at a distance d = 7 m from the goal line.
At to = 0, Messi received the ball at a point O at a distance D = 33 m from the goal line and he threw it with a velocity Vo making an angle a with the horizontal plane.
1. Using Newton's second law, Determine the equation of the trajectory of motion of the ball in terms of Vo, a and g.x.
2. When Navas saw the ball approached towards him, he raised his hand at the point A, and he jumped vertically upward trying to catch the ball, so their fingers reached a point E located at 2.95 m above the ground.
a- What should be the values of Vo and the angle a so that the ball passes through a point B 5 cm above the fingers of Navas and reaches the goal line?
b- Does the point B represent the maximum point reached by the ball? If Yes, justify your answer; If not, determine the maximum height reached by the ball.
c- Precise the instant at which the ball reaches the goal line.
d- Calculate the speed Vr of the ball when it reaches the goal line and deduce the angle that makes Vtr with the horizontal axis X'X.
3. At to 0 When Messi threw the ball, Sergio Ramos the defender of Real Madrid, which is located just next to Navas at a distance d from the goal line, runs without initial velocity towards the goal with an acceleration a=3.2 m/s² trying to prevent the goal. That's why he must reach the goal line before the ball, his motion is rectilinear along the (OX) axis.
a- Determine the time equation of motion of Sergio along the (OX) axis.
b- Does Messi score the goal and Barcelona wins or as usual the chance will be next to Real Madrid and Ramos prevent the goal? justify your answer. ​