Answer :
Answer:
38.66 degrees.
Explanation:
Here's how to calculate the angle of projection:
**Understanding the Relationships**
* **Range (R):** The horizontal distance the object travels.
* **Maximum Height (H):** The highest vertical point the object reaches.
* **Initial Velocity (u):** The velocity at which the object is launched.
* **Angle of Projection (θ):** The angle at which the object is launched from the horizontal.
**Formulas:**
* Range: R = (u² * sin(2θ)) / g
* Maximum Height: H = (u² * sin²θ) / (2g)
where g is the acceleration due to gravity (approximately 9.81 m/s²)
**Given Information:**
* R = 5H
* u = 200 m/s
**Calculation:**
1. **Substitute:** Since R = 5H, substitute this into the range equation:
5H = (u² * sin(2θ)) / g
2. **Equate:** Substitute the maximum height equation into the above equation:
5 * (u² * sin²θ) / (2g) = (u² * sin(2θ)) / g
3. **Simplify:** Cancel out common terms (u² and g) and simplify:
5 * sin²θ / 2 = sin(2θ)
4. **Double Angle Identity:** Use the double angle identity sin(2θ) = 2sinθcosθ:
5 * sin²θ / 2 = 2sinθcosθ
5. **Further Simplification:** Cancel out a sinθ term from both sides (assuming it's not zero):
5 * sinθ / 2 = 2cosθ
6. **Tangent:** Divide both sides by cosθ (assuming it's not zero) to get the tangent:
5 * tanθ / 2 = 2
7. **Solve for tanθ:** Isolate tanθ:
tanθ = 4/5
8. **Calculate θ:** Take the inverse tangent (arctan) of both sides:
θ = arctan(4/5)
9. **Approximate:** Use a calculator to find the approximate value:
θ ≈ 38.66°