An object at (0, 9) moves in straight-line two-dimensional motion to a position of (7, 15). After that, it has a displacement of 1 m in the negative  y
 -direction. Immediately after that, it moves in straight-line two-dimensional motion to a final position of (4, 16). Determine the distance traveled by the object going from its initial position to its final position. Express your answer rounded to 3 significant figures.

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.

Learn how to calculate the total distance traveled by an object in two-dimensional motion. This problem involves using the distance formula to compute distances between points and applying the Pythagorean theorem. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation