Addition of Vectors involving Two Dimensions

Sample problem

A hot-air balloon rises into the air and drifts with the wind at a rate of 24 km/h,40° E or N for 2 h. The wind shifts, so the balloon changes direction and drifts south at a rate of 40 km/h for 1.5 h before landing.

Determine the balloon's displacement for the flight.

To determine the balloon's displacement for the flight, we need to break the problem down into two parts: the first leg of the flight (drifting with the wind) and the second leg (drifting south). We'll use vector addition to solve the problem.

### 1. **First leg of the flight**:
- **Speed**: 24 km/h
- **Direction**: 40° east of north (40° E of N)
- **Time**: 2 hours

#### Step 1: Find the displacement for the first leg.
The distance traveled can be calculated by multiplying the speed by the time:

\[
\text{Distance}_1 = \text{Speed}_1 \times \text{Time}_1 = 24 \, \text{km/h} \times 2 \, \text{h} = 48 \, \text{km}
\]

Since the direction is 40° east of north, we need to break this displacement into its horizontal and vertical components:

- Horizontal component (east):  
\[
d_{1x} = 48 \, \text{km} \times \sin(40^\circ) \approx 48 \times 0.6428 = 30.86 \, \text{km}
\]

- Vertical component (north):  
\[
d_{1y} = 48 \, \text{km} \times \cos(40^\circ) \approx 48 \times 0.7660 = 36.77 \, \text{km}
\]

### 2. **Second leg of the flight**:
- **Speed**: 40 km/h
- **Direction**: South (180°)
- **Time**: 1.5 hours

#### Step 2: Find the displacement for the second leg.
The distance traveled is:

\[
\text{Distance}_2 = \text{Speed}_2 \times \text{Time}_2 = 40 \, \text{km/h} \times 1.5 \, \text{h} = 60 \, \text{km}
\]

Since this is directly south, the components are:
- Horizontal component:  
\[
d_{2x} = 0 \, \text{km}
\]
- Vertical component (south):  
\[
d_{2y} = -60 \, \text{km}
\]
(The negative sign indicates southward direction.)

### 3. **Determine total displacement**:
To find the total displacement, we sum the components from both legs.

- Total horizontal component:  
\[
d_x = d_{1x} + d_{2x} = 30.86 \, \text{km} + 0 \, \text{km} = 30.86 \, \text{km}
\]

- Total vertical component:  
\[
d_y = d_{1y} + d_{2y} = 36.77 \, \text{km} - 60 \, \text{km} = -23.23 \, \text{km}
\]

### 4. **Find the magnitude of the total displacement**:
The magnitude of the displacement is given by the Pythagorean theorem:

\[
\text{Displacement} = \sqrt{(d_x)^2 + (d_y)^2} = \sqrt{(30.86)^2 + (-23.23)^2}
\]

\[
\text{Displacement} = \sqrt{951.14 + 539.65} = \sqrt{1490.79} \approx 38.61 \, \text{km}
\]

### 5. **Find the direction of the displacement**:
The direction of the displacement can be found using the inverse tangent function:

\[
\theta = \tan^{-1} \left( \frac{|d_y|}{d_x} \right) = \tan^{-1} \left( \frac{23.23}{30.86} \right) \approx \tan^{-1}(0.753) \approx 37^\circ
\]

Since the resultant displacement is in the fourth quadrant (east and south), the angle is measured south of east.

### Final Answer:
- The magnitude of the balloon's displacement is approximately **38.61 km**.
- The direction is approximately **37° south of east**.

Would you like further details on any part of this solution or have questions?

---

Here are 5 related questions to consider:
1. How do you find the resultant of more than two vectors?
2. How would the calculation change if the balloon had traveled west instead of east?
3. What is the physical meaning of vector components in two dimensions?
4. How can the Pythagorean theorem be used for vectors in three dimensions?
5. What would the displacement be if the balloon drifted at varying speeds?

**Tip**: Always ensure angles are measured relative to the correct axis when breaking vectors into components.

A hot-air balloon rises into the air and drifts with the wind at a rate of 24 km/h, 40° E of N for 2 h. The wind shifts, so the balloon changes direction and drifts south at a rate of 40 km/h for 1.5 h before landing. Determine the balloon's displacement for the flight.

Learn how to calculate the displacement of a hot-air balloon using vector addition in two dimensions. This step-by-step solution covers trigonometry and the Pythagorean theorem to find the balloon's total displacement of 38.61 km at 37° south of east. Perfect for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation