Final answer:
To find out how long a ball booted at 27 m/s at a 40-degree angle above the horizontal takes to land, calculate the ball's vertical velocity component, apply the formula for time to peak height using the acceleration due to gravity, and double that time to get the total time in the air.
Step-by-step explanation:
To determine how long it takes for the soccer ball to land after being booted by Lionel Messi at 27 m/s at a 40-degree angle, we need to break down the initial velocity into its vertical and horizontal components using trigonometric functions. The vertical component (Vy) of the velocity can be found using Vy = V * sin(θ), and the horizontal component (Vx) is found with Vx = V * cos(θ). To find the time in the air, we will focus on the vertical motion because this is a symmetrical projectile motion problem where the ball lands at the same height from which it was kicked.
The time for the ball to reach its peak height can be calculated via the formula t = Vy / g, where g is the acceleration due to gravity (9.81 m/s² on Earth). When the initial velocity in the y-direction (upwards) is equal to the velocity at which the ball falls back down (which will be the same magnitude but opposite in direction), the total time will be double the time it took to reach the peak since the upward and downward journeys take the same amount of time. Hence, the total time the ball is in the air is T_total = 2 * (Vy / g).