Two motor boats A and B move from same point along a circle of radius 10 m in still water. The boats are so designed that they can move only with constant speeds. The boats A and B take 16 and 8 sec respectively to complete one circle in stationary water. Now water starts flowing at `t=0` with a speed `4(m)/(s)` in a fixed direction. Find the distance between the boats after `t=8` sec.

Options

A. 10 m
B. 20 m
C. 100 m
D. none of these

Explanation

To solve the problem, we need to analyze the motion of the two boats A and B as they move in a circular path while considering the effect of the water current.

### Step 1: Determine the speeds of the boats in still water.
- Boat A takes 16 seconds to complete one circle.
- Boat B takes 8 seconds to complete one circle.
- The circumference of the circle is given by \( C = 2\pi r \), where \( r = 10 \) m.

Calculating the circumference:
\[ C = 2\pi \times 10 = 20\pi \text{ m} \]

Now, we can find the speeds of both boats:
- Speed of boat A (\( v_A \)):
\[ v_A = \frac{C}{\text{time}} = \frac{20\pi}{16} = \frac{5\pi}{4} \text{ m/s} \]
- Speed of boat B (\( v_B \)):
\[ v_B = \frac{C}{\text{time}} = \frac{20\pi}{8} = \frac{5\pi}{2} \text{ m/s} \]

### Step 2: Analyze the effect of the water current.
The water starts flowing at \( t = 0 \) with a speed of \( 4 \text{ m/s} \) in a fixed direction. This current will affect both boats equally since they are moving in the same direction.

The effective speeds of the boats in the direction of the current are:
- Effective speed of boat A (\( v_A' \)):
\[ v_A' = v_A + 4 = \frac{5\pi}{4} + 4 \]
- Effective speed of boat B (\( v_B' \)):
\[ v_B' = v_B + 4 = \frac{5\pi}{2} + 4 \]

### Step 3: Determine the positions of the boats after 8 seconds.
Now we calculate the distance each boat travels in 8 seconds:
- Distance traveled by boat A in 8 seconds:
\[ d_A = v_A' \times 8 = \left(\frac{5\pi}{4} + 4\right) \times 8 \]
- Distance traveled by boat B in 8 seconds:
\[ d_B = v_B' \times 8 = \left(\frac{5\pi}{2} + 4\right) \times 8 \]

### Step 4: Find the angular positions of the boats.
Since both boats are moving in a circle, we need to find their angular positions:
- The angular distance covered by boat A:
\[ \theta_A = \frac{d_A}{r} = \frac{d_A}{10} \]
- The angular distance covered by boat B:
\[ \theta_B = \frac{d_B}{r} = \frac{d_B}{10} \]

### Step 5: Calculate the distance between the boats.
After 8 seconds, the distance between the two boats can be calculated using the formula:
\[ \text{Distance} = 2r \sin\left(\frac{\theta_B - \theta_A}{2}\right) \]

### Step 6: Substitute and calculate.
After substituting the values and calculating, we find that the distance between the two boats after 8 seconds is:
\[ \text{Distance} = 20 \text{ m} \]

### Final Answer:
The distance between the boats after \( t = 8 \) seconds is \( 20 \text{ m} \).

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