A soccer ball is kicked from the ground with an initial speed of 18.9 m/s at an upward angle of 47.0˚. A player 54.4 m away in the direction of the kick starts running to meet the ball at that instant. What must be his average speed if he is to meet the ball just before it hits the ground? Neglect air resistance.

Vo = 18.9m/s[47o]

Xo = 18.9*cos47 = 12.9 m/s.

Range = Vo^2*sin(2A)/g
Range = 18.9^2*sin94/9.8 = 36.4 m.

Range = Xo * t = 36.4 m.
12.9 * t = 36.4
t = 2.82 s. in flight.

d = 54.4-36.4 = 18 m To run in 2.82 s.

Speed = d/t = 18m/2.82s = 6.38 m/s.

To solve this problem, we need to find the time it takes for the soccer ball to hit the ground after being kicked, and then use that time to calculate the required average speed of the player.

Step 1: Find the time it takes for the soccer ball to hit the ground
We can break down the initial velocity of the soccer ball into its horizontal and vertical components.

Horizontal component (Vx): Vx = V * cos(θ)
where V is the initial speed of the soccer ball and θ is the angle of the kick.

Vertical component (Vy): Vy = V * sin(θ)

Since we're neglecting air resistance, the time it takes for the ball to reach the highest point of its trajectory (when Vy = 0) is equal to the time it takes for the ball to fall back to the ground.

Using the equation for vertical displacement under constant acceleration:
h = (1/2) * g * t^2
where h is the maximum height, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time.

At the maximum height, the vertical velocity Vy becomes 0:
0 = Vy - g * t
t = Vy / g

Since Vy = V * sin(θ), we can substitute it in the equation:
t = (V * sin(θ)) / g

The total time it takes for the ball to hit the ground is twice this time because it takes the same amount of time for the ball to reach the maximum height as it does to fall to the ground:
total_time = 2 * t

Step 2: Calculate the required average speed of the player
The player needs to cover a distance of 54.4 m in the same amount of time it takes for the ball to hit the ground.

average_speed = total_distance / total_time
where total_distance is the distance between the player and the starting point.

Substitute the values:
average_speed = 54.4 m / (2 * t)

Now, let's calculate the solution using the given values:
V = 18.9 m/s (initial speed)
θ = 47.0° (angle of the kick)
g = 9.8 m/s^2 (acceleration due to gravity)
total_distance = 54.4 m

1. Calculate t:
t = (V * sin(θ)) / g
t = (18.9 m/s * sin(47.0°)) / 9.8 m/s^2

2. Calculate total_time:
total_time = 2 * t

3. Calculate average_speed:
average_speed = 54.4 m / (2 * t)

Plug in the values to find the solution.