To find the distance the soccer ball travels through the air, we need to analyze the motion in the horizontal and vertical directions separately. Let's assume there is no air resistance.
First, we'll find the initial horizontal velocity (Vx) and initial vertical velocity (Vy) of the ball using the given initial velocity and launch angle.
Vx = initial velocity * cos(angle) Vy = initial velocity * sin(angle)
Given: Initial velocity = 30 mph Angle = 15 degrees
Converting the initial velocity to feet per second (fps):
Initial velocity = 30 mph * (5280 ft/mile) / (3600 s/hour) ≈ 44 ft/s
Now, we can calculate the initial horizontal and vertical velocities:
Vx = 44 ft/s * cos(15 degrees) Vy = 44 ft/s * sin(15 degrees)
Calculating Vx and Vy:
Vx ≈ 42.496 ft/s Vy ≈ 11.788 ft/s
The horizontal motion of the ball is unaffected by gravity, so we can determine the time it spends in the air using the horizontal velocity (Vx) and the total horizontal distance traveled (D):
D = Vx * time
We need to find the time of flight, which is the total time the ball is in the air. The time it takes for the ball to reach its maximum height is equal to the time it takes for the ball to fall back down to the ground. This can be calculated using the vertical motion.
In the vertical direction, we know the initial vertical velocity (Vy), the acceleration due to gravity (g ≈ 32.2 ft/s²), and we need to find the time it takes for the ball to reach its maximum height (t_max).
Using the equation for vertical displacement:
Vy * t_max - (1/2) * g * t_max² = 0
Simplifying the equation:
t_max = Vy / (g/2)
Calculating t_max:
t_max ≈ (11.788 ft/s) / (32.2 ft/s²/2) ≈ 0.3667 s
Since the total time of flight is twice the time it takes to reach the maximum height (t_max), we have:
Total time of flight = 2 * t_max
Total time of flight ≈ 2 * 0.3667 s ≈ 0.7334 s
Now we can calculate the horizontal distance traveled by the ball:
D = Vx * time
D ≈ 42.496 ft/s * 0.7334 s ≈ 31.153 ft
Therefore, the soccer ball travels approximately 31.153 feet through the air.