Trigonometry Case Study - Hot Air Balloon Festival
Case Study 2: The Hot Air Balloon Festival
Explore how angles of elevation change as the balloon moves away and calculate real-world distances using trigonometry
๐ Jaipur, India | ๐ช Festival Setting | ✈️ Motion Analysis
The Scenario
๐ Setting: Hot Air Balloon Festival in Jaipur
During a vibrant hot air balloon festival, a girl observes a magnificent hot air balloon floating gracefully in the sky. The balloon maintains a constant altitude as it moves horizontally across the landscape.
๐ First Observation:
Angle of elevation = 60°
⏱️ Later Observation:
Angle of elevation = 30°
๐ Constant Height:
80√3 m from the ground
๐ Position 1
Balloon is closer to the girl. Angle of elevation is larger (60°)
➡️ Movement
Balloon moves horizontally at constant height
๐ Position 2
Balloon moves away. Angle of elevation becomes smaller (30°)
Question 1: Draw a Labeled Diagram
Draw a rough labeled diagram representing the situation.
✓ Solution
Key Elements in the Diagram:
- Point A: Observer's position (girl's eye level)
- Point B: Initial balloon position (angle = 60°)
- Point C: Final balloon position (angle = 30°)
- Height h: Constant altitude = 80√3 m
- Dashed lines: Lines of sight at different angles
- Horizontal line: Reference line through observer
Question 2: Understanding Angle Change
Why does the angle of elevation decrease as the balloon moves away?
✓ Solution with Explanation
๐ Geometric Principle
The angle of elevation is the angle between the horizontal line and the line of sight to the object. As the balloon moves away horizontally at constant height, the line of sight becomes more and more horizontal.
๐ Closer Distance
When horizontal distance is small, the line of sight makes a large angle with the horizontal.
Angle = 60° (Steep)
➡️ Increased Distance
When horizontal distance is large, the line of sight makes a small angle with the horizontal.
Angle = 30° (Shallow)
๐ข Mathematical Proof
For angle of elevation ฮธ:
tan ฮธ = h / d
where h = height (constant) and d = horizontal distance
When d increases → tan ฮธ decreases → ฮธ decreases
Example: If d = 100 m, then tan ฮธ = 80√3/100
If d = 240 m, then tan ฮธ = 80√3/240 (much smaller angle)
๐ก Real-Life Analogy
Imagine watching a bird fly upward from your position. When it's close, you look up steeply (large angle). As it moves away horizontally, you look less steeply (smaller angle) even though it maintains the same height.
๐ Inverse Relationship
Distance ↑ ⟹ Angle ↓
Distance ↓ ⟹ Angle ↑
Question 3: Calculate Horizontal Distance
Calculate the horizontal distance covered by the balloon during the interval.
✓ Step-by-Step Solution
Identify Given Information
• Height of balloon = h = 80√3 m (constant)
• Initial angle of elevation = ฮธ₁ = 60°
• Final angle of elevation = ฮธ₂ = 30°
• We need to find: Distance covered = d₂ - d₁
Set Up Trigonometric Equations
For Position 1 (60° angle):
tan 60° = h / d₁
√3 = 80√3 / d₁
d₁ = 80√3 / √3 = 80 m
Calculate Distance for Position 2
For Position 2 (30° angle):
tan 30° = h / d₂
1/√3 = 80√3 / d₂
d₂ = 80√3 × √3
d₂ = 80 × 3 = 240 m
Calculate Total Distance Covered
Distance covered by balloon = d₂ - d₁
= 240 - 80
= 160 m
๐ Horizontal Distance Covered
160 m
The balloon travels 160 meters horizontally while maintaining constant altitude
✓ Verification
• d₁ = 80 m → tan⁻¹(80√3/80) = tan⁻¹(√3) = 60° ✓
• d₂ = 240 m → tan⁻¹(80√3/240) = tan⁻¹(1/√3) = 30° ✓
• Difference = 240 - 80 = 160 m ✓
Alternative Question: Calculate Speed
If the speed of the balloon was constant, and it took 10 seconds to cover that distance, find the speed of the balloon in m/s.
✓ Step-by-Step Solution
Recall Distance from Previous Question
From Question 3, we calculated:
Distance = 160 m
Time taken = 10 seconds
Apply Speed Formula
Speed = Distance / Time
Speed = 160 / 10
Calculate Speed in m/s
Speed = 16 m/s
This is the constant speed at which the balloon moves horizontally
Bonus: Convert to km/h
Speed in km/h = 16 × (18/5)
= 16 × 3.6
= 57.6 km/h
⚡ Speed of the Balloon
16 m/s
or
57.6 km/h
A reasonable speed for a hot air balloon in motion
๐ Real-World Context
Typical Hot Air Balloon Speeds:
- • Hovering/Stationary: 0 m/s
- • Drifting slowly: 2-5 m/s
- • Active flight: 5-15 m/s
- • Our balloon: 16 m/s (quite fast!)
- • Maximum typical: ~20 m/s
Complete Summary & Formula Reference
๐ Trigonometric Ratio
tan ฮธ = opposite / adjacent
tan ฮธ = height / distance
⚡ Speed Formula
Speed = Distance / Time
v = d / t
| Angle | sin ฮธ | cos ฮธ | tan ฮธ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
๐ Question Answers
Q1: Draw Diagram
Labeled with A, B, C points ✓
Q2: Angle Decrease Reason
Distance ↑ → Angle ↓
Q3: Distance
160 m
Q4: Speed
16 m/s (57.6 km/h)
๐ก Key Takeaways
- ✓ Angle of elevation depends on observer's position
- ✓ Greater distance = Smaller angle
- ✓ Height remains constant in this problem
- ✓ Trigonometry solves real-world motion problems
- ✓ Speed calculation combines distance and time
No comments