To solve this problem, we need to calculate the total distance traveled by the object as it moves through its specified path. We'll break the motion into individual segments, calculate the distance for each segment, and then sum them up.

Step 1: Calculate the distance between the initial position $(0, 9)$ and the position $(7, 15)$

The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Using the given points $(0, 9)$ and $(7, 15)$:

$\text{Distance}_1 = \sqrt{(7 - 0)^2 + (15 - 9)^2} = \sqrt{7^2 + 6^2} = \sqrt{49 + 36} = \sqrt{85} \approx 9.22 \, \text{m}$

Step 2: Calculate the distance of the displacement in the negative $y$-direction

The object moves 1 meter in the negative $y$-direction, so:

$\text{Distance}_2 = 1 \, \text{m}$

Step 3: Calculate the distance between the position after the $y$-displacement $(7, 14)$ and the final position $(4, 16)$

Using the points $(7, 14)$ and $(4, 16)$:

$\text{Distance}_3 = \sqrt{(4 - 7)^2 + (16 - 14)^2} = \sqrt{(-3)^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.61 \, \text{m}$

Step 4: Sum the distances to find the total distance traveled

Now, add up the distances from each segment:

$\text{Total Distance} = \text{Distance}_1 + \text{Distance}_2 + \text{Distance}_3 \approx 9.22 + 1 + 3.61 \approx 13.8 \, \text{m}$

So, the total distance traveled by the object is approximately 13.8 meters.

Would you like further details or have any questions?

Here are some related questions:

1. How would the distance change if the object moved diagonally after the $y$-displacement?
2. What is the significance of displacement compared to distance traveled in physics?
3. How would you calculate the total displacement for this motion?
4. How does the Pythagorean theorem relate to finding distances in two dimensions?
5. How would the distance calculation change in a three-dimensional space?

Tip: When dealing with two-dimensional motion, breaking the problem into segments can simplify the calculations.