A soccer ball is kicked from the ground with an initial speed of 18.0 m/s at an upward angle of 41.2˚. A player 51.1 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.

To find the player's average speed, we need to determine the time it takes for the soccer ball to hit the ground. Then, since the player starts running at that instant, we can calculate the average speed as the distance traveled by the player divided by the time.

Let's break down the problem step by step:

Step 1: Find the time it takes for the soccer ball to reach the ground.
Since we neglect air resistance, we can analyze the motion of the ball in the vertical direction. We can use the equation:
y = y0 + v0y * t - (1/2) * g * t^2
where:
y = 0 (since the ball hits the ground)
y0 = 0 (initial height)
v0y = v0 * sin(theta) (vertical component of initial velocity)
g = 9.8 m/s^2 (acceleration due to gravity)
t = time

Plugging in the values:
0 = 0 + (18.0 m/s) * sin(41.2°) * t - (1/2) * (9.8 m/s^2) * t^2

Simplifying the equation and solving for t will give us the time it takes for the ball to reach the ground.

Step 2: Calculate the distance traveled by the player.
By the time the player starts running, the soccer ball has traveled a horizontal distance of 51.1 m, given in the problem. This is the distance the player needs to cover to meet the ball just before it hits the ground.

Step 3: Find the average speed of the player.
Average speed is calculated as the total distance traveled divided by the total time taken. In this case, the distance traveled by the player is 51.1 m (from Step 2), and the time is the result we obtained in Step 1.

Average speed = distance traveled / time

Calculate the average speed using the values obtained in the previous steps, and the final result will give us the answer to the problem.