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A boat moves relative to water with a ve...

A boat moves relative to water with a velocity which is 'n' times the river flow
a) If `n lt 1` boat can not cross the river
b) If n = 1 boat can not cross the river without drifting
c) If `n gt 1` boat can cross the river along shortest path
d) Boat can cross the river what ever is the value of n excluding zero value

A

only a is correct

B

a, b are correct

C

c, d are correct

D

b, c & d are correct

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the motion of the boat relative to the river flow based on the value of 'n', which represents the ratio of the boat's velocity to the river's velocity. ### Step-by-Step Solution: 1. **Define Variables:** - Let \( v_0 \) be the velocity of the river. - The velocity of the boat relative to water is \( n \cdot v_0 \). 2. **Case when \( n < 1 \):** - If \( n < 1 \), the velocity of the boat \( (n \cdot v_0) \) is less than the velocity of the river \( (v_0) \). - This means the river's flow is stronger than the boat's ability to move across the river. Hence, the boat cannot cross the river at all. - Conclusion: **Statement (a) is correct.** 3. **Case when \( n = 1 \):** - If \( n = 1 \), the boat's velocity is equal to the river's velocity. - The boat can move directly across the river, but it will drift downstream with the current. Therefore, it cannot reach the point directly across from its starting point without drifting. - Conclusion: **Statement (b) is correct.** 4. **Case when \( n > 1 \):** - If \( n > 1 \), the boat's velocity is greater than the river's velocity. - The boat can now move across the river while compensating for the river's flow. It can achieve a resultant velocity that allows it to cross directly to the opposite bank without drifting. - Conclusion: **Statement (c) is correct.** 5. **General Case for Any \( n \neq 0 \):** - Regardless of the value of \( n \) (as long as \( n \) is not zero), the boat will always have some velocity relative to the water. This means it can always attempt to cross the river, even if it cannot do so effectively (as in the case of \( n < 1 \)). - Conclusion: **Statement (d) is correct.** ### Final Conclusion: - The correct statements are: **(b), (c), and (d)**. - Therefore, the correct option is **d**: "Boat can cross the river whatever is the value of n excluding zero value."
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