Final answer:
Using the principles of projectile motion and assuming no air resistance, the initial speed required for the soccer ball to just pass over the goal 2.4 m above the ground, given a kicking angle of 40° and a distance of 30 m, is approximately 25 m/s.
Step-by-step explanation:
To find the initial speed of the soccer ball, we can use the equations for projectile motion. As we ignore air resistance, the motion can be analyzed separately in horizontal and vertical directions. Given the initial angle θ = 40°, distance to the goal (range) R = 30 m, and the height of the goal H = 2.4 m, we start by using the equation for the vertical motion:
H = V0y·t - (1/2)g·t2
where V0y is the initial vertical speed, t is the time of flight, and g is the acceleration due to gravity (approximately 9.81 m/s2). Since we're looking for V0, the total initial speed, we have to find V0y first and then use it to calculate V0.
V0y can be found by using the formula V0y = V0·sin(θ), and the time of flight (t) can be found by dividing the total horizontal distance (R) by the initial horizontal speed (V0x), which is V0x = V0·cos(θ).
After calculating the initial vertical and horizontal speeds separately, we use the Pythagorean theorem to find the resultant initial velocity (V0).
The correct initial speed of the ball that would just pass over the goal would be approximately 25 m/s, which corresponds to option b).