Final answer:
To find the initial speed of the soccer ball, calculate the vertical component of velocity (Viy) after solving the equation of motion for vertical fall, and then apply trigonometric relations to find the overall initial speed (Vi).
Step-by-step explanation:
To determine the initial speed of a soccer ball kicked at an angle of 73.0° above horizontal and in the air for 4.80 seconds, we consider the vertical motion of the ball. Using the equation of motion for constant acceleration (which is gravity in this case), we have 0 = Viy * t + (1/2) * g * t^2, where Viy is the initial vertical speed, t is the time the ball is in the air, and g is the acceleration due to gravity. Solving for Viy, we can determine the initial vertical component of the velocity.
The initial speed can then be found by using the trigonometric relations in which the Viy is equal to the initial speed Vi times the sine of the launch angle. Thus, Viy = Vi * sin(θ). The initial speed Vi can then be calculated as Vi = Viy / sin(θ). Given that the ball is in the air for 4.80 s, we assume that the total time for the upward and downward journey is equal. Therefore, t can be taken as half the total time the ball is airborne. By using the acceleration due to gravity (g = 9.81 m/s²), we can calculate the initial vertical speed. Subsequently, using the angle of 73.0°, we can calculate the initial speed of the soccer ball. Since air resistance is ignored, this approach provides us with an appropriate approximation of the initial speed needed for such a soccer ball kick.