Two bodies are thrown with the same initial velocity of 30 m/s. One at 17°, other at 73° to the horizontal. The sum of the maximum heights reached by them is [g = 10 `m//s^(2)`]

Options

A. 45 m
B. 450 m
C. 4.5 m
D. 20 m

Answer

To solve the problem, we need to calculate the maximum heights reached by two bodies thrown at angles of 17° and 73° with the same initial velocity of 30 m/s. We will use the formula for maximum height in projectile motion:

\[ H = \frac{u^2 \sin^2 \theta}{2g} \]

where:
- \( H \) is the maximum height,
- \( u \) is the initial velocity,
- \( \theta \) is the angle of projection,
- \( g \) is the acceleration due to gravity.

### Step 1: Calculate the maximum height for the first body (θ = 17°)

1. **Identify the values:**
- Initial velocity \( u = 30 \, \text{m/s} \)
- Angle \( \theta_1 = 17° \)
- Acceleration due to gravity \( g = 10 \, \text{m/s}^2 \)

2. **Calculate \( \sin(17°) \):**
- Using a calculator, \( \sin(17°) \approx 0.2924 \)

3. **Calculate \( \sin^2(17°) \):**
- \( \sin^2(17°) \approx (0.2924)^2 \approx 0.0855 \)

4. **Substitute into the height formula:**
\[ H_1 = \frac{(30)^2 \cdot 0.0855}{2 \cdot 10} \]
\[ H_1 = \frac{900 \cdot 0.0855}{20} \]
\[ H_1 = \frac{76.95}{20} \approx 3.85 \, \text{m} \]

### Step 2: Calculate the maximum height for the second body (θ = 73°)

1. **Identify the values:**
- Angle \( \theta_2 = 73° \)

2. **Calculate \( \sin(73°) \):**
- Using a calculator, \( \sin(73°) \approx 0.9563 \)

3. **Calculate \( \sin^2(73°) \):**
- \( \sin^2(73°) \approx (0.9563)^2 \approx 0.9145 \)

4. **Substitute into the height formula:**
\[ H_2 = \frac{(30)^2 \cdot 0.9145}{2 \cdot 10} \]
\[ H_2 = \frac{900 \cdot 0.9145}{20} \]
\[ H_2 = \frac{823.05}{20} \approx 41.15 \, \text{m} \]

### Step 3: Calculate the sum of the maximum heights

1. **Add the two heights:**
\[ H = H_1 + H_2 \]
\[ H = 3.85 + 41.15 = 45 \, \text{m} \]

### Final Answer
The sum of the maximum heights reached by the two bodies is approximately **45 m**.

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