A man is sitting on the shore of a river. He is in the line of a 1.0 m long boat and is 5.5 m away from the centre of the boat. He sishes to throw an apple into the boat. If he can throuw the apple only wihta speed of 10 m/s, find the minimum and maximum angles of projection for successful shot. Assume that the point of projection and the edge of the boat are in the same horizontal level

Explanation

To solve the problem of determining the minimum and maximum angles of projection for the man to successfully throw an apple into the boat, we can follow these steps:

### Step 1: Understand the Problem
The man is 5.5 m away from the center of the boat, which is 1 m long. Therefore, the distance from the man to the nearest edge of the boat is \(5.5 - 0.5 = 5.0\) m, and the distance to the far edge of the boat is \(5.5 + 0.5 = 6.0\) m.

### Step 2: Use the Range Formula
The range \(R\) of a projectile is given by the formula:
\[ R = \frac{u^2 \sin 2\theta}{g} \]
where:
- \(u\) = initial velocity (10 m/s)
- \(g\) = acceleration due to gravity (approximately \(10 \, \text{m/s}^2\))
- \(\theta\) = angle of projection

### Step 3: Calculate for the Minimum Distance (5.0 m)
To find the angle for the minimum distance (5.0 m):
\[ 5.0 = \frac{(10)^2 \sin 2\theta_1}{10} \]
This simplifies to:
\[ 5.0 = 10 \sin 2\theta_1 \]
\[ \sin 2\theta_1 = 0.5 \]
From this, we can find \(2\theta_1\):
\[ 2\theta_1 = 30^\circ \quad \text{or} \quad 2\theta_1 = 150^\circ \]
Thus, the possible angles are:
\[ \theta_1 = 15^\circ \quad \text{or} \quad \theta_1 = 75^\circ \]

### Step 4: Calculate for the Maximum Distance (6.0 m)
Now, for the maximum distance (6.0 m):
\[ 6.0 = \frac{(10)^2 \sin 2\theta_2}{10} \]
This simplifies to:
\[ 6.0 = 10 \sin 2\theta_2 \]
\[ \sin 2\theta_2 = 0.6 \]
From this, we can find \(2\theta_2\):
\[ 2\theta_2 = \arcsin(0.6) \quad \text{or} \quad 2\theta_2 = 180^\circ - \arcsin(0.6) \]
Calculating \(\arcsin(0.6)\):
\[ 2\theta_2 \approx 36.87^\circ \quad \text{or} \quad 2\theta_2 \approx 143.13^\circ \]
Thus, the possible angles are:
\[ \theta_2 \approx 18.43^\circ \quad \text{or} \quad \theta_2 \approx 71.57^\circ \]

### Step 5: Conclusion
The minimum angle of projection is \(15^\circ\) and the maximum angle of projection is \(75^\circ\).

### Final Answer
- Minimum angle: \(15^\circ\)
- Maximum angle: \(75^\circ\)