To determine the ball's total displacement as it travels between Emily and Luisa, we need to consider the movements in terms of vectors and use the Pythagorean theorem.

1. Emily passes the ball 6.0 meters directly across the field to Kara (horizontal movement).
2. Kara then kicks the ball 14.5 meters directly down the field to Luisa (vertical movement).

These movements form a right triangle with the following legs:

• Horizontal leg = 6.0 meters
• Vertical leg = 14.5 meters

The total displacement is the hypotenuse of this right triangle. We use the Pythagorean theorem to find it:

$\text{Displacement} = \sqrt{(6.0 \, \text{m})^2 + (14.5 \, \text{m})^2}$

First, we calculate the squares of the legs:

$(6.0 \, \text{m})^2 = 36.0 \, \text{m}^2$

$(14.5 \, \text{m})^2 = 210.25 \, \text{m}^2$

Now, we add these values:

$36.0 \, \text{m}^2 + 210.25 \, \text{m}^2 = 246.25 \, \text{m}^2$

Finally, we take the square root of the sum to find the displacement:

$\text{Displacement} = \sqrt{246.25 \, \text{m}^2} \approx 15.7 \, \text{m}$

So, the ball’s total displacement as it travels between Emily and Luisa is approximately 15.7 meters.

Do you want any further details or have any questions?

Here are some related questions you might want to ask:

1. How can we find the angle of the total displacement in this problem?
2. What are some other examples of using the Pythagorean theorem in real-life scenarios?
3. How does vector addition help in finding total displacement?
4. What is the difference between distance and displacement?
5. Can we use trigonometry to solve similar problems involving angles and lengths?

Tip: When solving for total displacement involving multiple movements, always break down the problem into right triangles and apply the Pythagorean theorem to simplify the calculations.