child releases a balloon at a park. The balloon travels up into the air 8.20 m,
and east across the park a distance of 23.0 m, before getting stuck in a tree. Draw
a diagram representing this situation. What is the resultant displacement of the
balloon? Express your answer in components, and then determine the magnitude.

To solve the problem, let's break down the situation:

1. The balloon travels **upward** vertically by 8.20 m.
2. It also moves **eastward** horizontally by 23.0 m.

This forms a right triangle, where:
- The vertical leg represents the upward displacement: \( 8.20 \, \text{m} \).
- The horizontal leg represents the eastward displacement: \( 23.0 \, \text{m} \).

We need to:
1. **Determine the components of the displacement:**
   - The vertical displacement is \( 8.20 \, \text{m} \) in the positive \( y \)-direction.
   - The horizontal displacement is \( 23.0 \, \text{m} \) in the positive \( x \)-direction.

   Hence, the displacement components are:
   \[
   \vec{d} = (23.0 \hat{i} + 8.20 \hat{j}) \, \text{m}
   \]
   where \( \hat{i} \) and \( \hat{j} \) are the unit vectors in the \( x \)- and \( y \)-directions, respectively.

2. **Calculate the magnitude of the resultant displacement** using the Pythagorean theorem:
   \[
   \text{Magnitude of displacement} = \sqrt{(23.0)^2 + (8.20)^2}
   \]

Let's calculate the magnitude.The magnitude of the resultant displacement is approximately **24.42 meters**.

To summarize:
- The displacement components are: \( (23.0 \hat{i} + 8.20 \hat{j}) \, \text{m} \).
- The magnitude of the resultant displacement is \( 24.42 \, \text{m} \).

Would you like further details or have any questions about the process?

Here are some related questions for you to explore:
1. What angle does the resultant displacement make with the horizontal?
2. How would the displacement change if the balloon traveled twice as high?
3. What would happen if the balloon also moved northward?
4. How could you represent this scenario in three dimensions?
5. What is the difference between displacement and total distance traveled?

**Tip:** Displacement is a vector quantity, meaning it includes both magnitude and direction, unlike distance which is scalar and only reflects the path length.

A child releases a balloon at a park. The balloon travels up into the air 8.20 m, and east across the park a distance of 23.0 m, before getting stuck in a tree. Draw a diagram representing this situation. What is the resultant displacement of the balloon? Express your answer in components, and then determine the magnitude.

Learn how to calculate the resultant displacement of a balloon that travels vertically 8.20 m and horizontally 23.0 m. This problem involves key vector concepts, such as displacement components and the Pythagorean theorem. Suitable for students in Grades 9-11.

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