Final answer:
The ball kicked by a goalie at a 20 m/s speed at a 60-degree angle will stay in the air for approximately 3.48 seconds, considering only the vertical motion and gravity. None of the provided choices match this calculation, suggesting a possible typo or error in the question or choices.
Step-by-step explanation:
The question pertains to the time a soccer ball stays in the air after being kicked by a goalie. This scenario can be analyzed using the concepts of projectile motion. We will calculate the time the ball was in the air after being kicked at a 20 m/s speed at a 60-degree angle. To find the duration of the flight, we need to consider only the vertical component of the initial velocity, since the time the ball spends in the air is determined by the time it takes to rise and fall back to the ground level, influenced only by gravity.
To find the vertical component of the initial velocity, we use sin(60 degrees) = √3/2. Therefore, the vertical component of the velocity (Vy) is (20 m/s) √3/2. Plugging in the values:
Vy = (20 m/s) (√3/2) = 10√3 m/s.
Since gravity acts downward at 9.81 m/s² (on Earth), the time (t) for the ball to reach the peak of its trajectory is given by:
t = Vy / g
t = (10√3 m/s) / (9.81 m/s²)
t ≈ 1.74 seconds.
The total time in the air will be double that figure because the time to go up is the same as the time to come down. Hence, the total time (T) the ball spends in the air is:
T = 2t
T = 2 × 1.74 seconds ≈ 3.48 seconds.
However, this result does not correspond to any of the provided answer choices. It is possible there may have been a miscalculation or typo in the question or in the available choices. Thus, none of the answer choices (A) 1.34 seconds, (B) 2.68 seconds, (C) 0.67 seconds, (D) 3.01 seconds, is correct based on the given conditions and calculations.