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°

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