To determine the total time a ball is in the sky, we need to consider the motion of the ball in the vertical direction. We can break down the initial velocity of the ball into horizontal and vertical components.
Given:
Initial speed of the ball (v) = 39 m/s
Launch angle (θ) = 72 degrees
First, we need to determine the vertical component of the initial velocity (vy) using trigonometry:
vy = v * sin(θ)
Substituting the given values:
vy = 39 m/s * sin(72 degrees)
vy ≈ 37.25 m/s (rounded to two decimal places)
Now, we can calculate the time taken for the ball to reach the highest point of its trajectory. At the highest point, the vertical component of the velocity will be zero. We can use the following equation of motion:
vy = u + at
Since the ball is at its highest point, the final velocity (v) is zero, the initial velocity (u) is vy, and the acceleration (a) is due to gravity (approximately -9.8 m/s^2):
0 = vy + (-9.8 m/s^2) * t_max
Solving for t_max:
t_max = vy / 9.8 m/s^2
t_max ≈ 3.80 seconds (rounded to two decimal places)
Next, we can calculate the total time of flight by considering the time taken to reach the highest point and doubling it:
Total time of flight = 2 * t_max
Total time of flight ≈ 2 * 3.80 seconds
Total time of flight ≈ 7.60 seconds (rounded to two decimal places)
Therefore, the total time the ball is in the sky is approximately 7.60 seconds.