Final answer:
To determine how far the ball travels horizontally, we break down the initial velocity into horizontal and vertical components using trigonometry, then calculate the time the ball spends in the air, and finally multiply the horizontal velocity by this time to find the horizontal distance, which is approximately 30.96 meters.
Step-by-step explanation:
To calculate how far the football travels horizontally, we will use the principles of projectile motion. Given that the velocity of the ball is 18.0 m/s and the angle of projection is 35.0 degrees, we first need to find the horizontal component of the velocity (Vx). We can use the cosine function to isolate this component:
Vx = V * cos(θ) = 18.0 m/s * cos(35.0°) = 18.0 m/s * 0.8192 = 14.746 m/s
Now, to find the time (t) the ball stays in the air, we need the vertical component of the velocity (Vy) and the acceleration due to gravity (g = 9.81 m/s2). From the given angle, Vy can be calculated as:
Vy = V * sin(θ) = 18.0 m/s * sin(35.0°) = 18.0 m/s * 0.5736 = 10.325 m/s
Since the ball is caught at the same height it was kicked, the time to rise to the maximum height equals the time to fall back down. We calculate the time to rise to the maximum height (tup) using Vy and g. Keeping in mind that at maximum height, the vertical velocity is 0 m/s:
tup = Vy / g = 10.325 m/s / 9.81 m/s2 ≈ 1.05 s
Therefore, the total time in the air is twice the time it takes to reach maximum height:
t = 2 * tup ≈ 2 * 1.05 s ≈ 2.10 s
Finally, to find the horizontal distance (range) the ball travels, we multiply the time by the horizontal velocity:
Range = Vx * t = 14.746 m/s * 2.10 s ≈ 30.96 m
The ball travels approximately 30.96 meters horizontally.