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A man wants to cross the river to an exactly opposite point on the other bank if he can row his boat with root two times the velocity of water current at what angle to the current must be keep the word moving?
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A man wants to cross the river to an exactly opposite point on the oth...
Introduction
To cross a river directly to the opposite point while accounting for the current, the man must row at an angle. The challenge lies in compensating for the water’s velocity.
Understanding the Velocities
- The velocity of the man’s rowing (Vb) = √2 * Vw (where Vw is the current's velocity).
- To reach the opposite bank, the resultant velocity must have no downstream component.
Breaking Down the Problem
- The current pushes downstream, affecting the man's path.
- The rower must create a component of his rowing velocity that counteracts the current.
Key Angles and Components
- Let θ be the angle between the rowing direction and the current.
- The rowing velocity can be divided into two components:
- Upstream Component: Vb * sin(θ) (this counters the current)
- Cross-River Component: Vb * cos(θ) (this moves him across the river)
Setting Up the Equation
- To cross directly:
- Upstream Component = Current Velocity
- Vb * sin(θ) = Vw
- Substituting Vb:
- √2 * Vw * sin(θ) = Vw
- Simplifying gives:
- sin(θ) = 1/√2
- Therefore, θ = 45 degrees
Conclusion
To cross the river to an exact point directly opposite, the man must row at an angle of 45 degrees upstream relative to the current. This angle allows him to effectively counteract the current while maintaining a straight path across the river.
To cross a river directly to the opposite point while accounting for the current, the man must row at an angle. The challenge lies in compensating for the water’s velocity.
Understanding the Velocities
- The velocity of the man’s rowing (Vb) = √2 * Vw (where Vw is the current's velocity).
- To reach the opposite bank, the resultant velocity must have no downstream component.
Breaking Down the Problem
- The current pushes downstream, affecting the man's path.
- The rower must create a component of his rowing velocity that counteracts the current.
Key Angles and Components
- Let θ be the angle between the rowing direction and the current.
- The rowing velocity can be divided into two components:
- Upstream Component: Vb * sin(θ) (this counters the current)
- Cross-River Component: Vb * cos(θ) (this moves him across the river)
Setting Up the Equation
- To cross directly:
- Upstream Component = Current Velocity
- Vb * sin(θ) = Vw
- Substituting Vb:
- √2 * Vw * sin(θ) = Vw
- Simplifying gives:
- sin(θ) = 1/√2
- Therefore, θ = 45 degrees
Conclusion
To cross the river to an exact point directly opposite, the man must row at an angle of 45 degrees upstream relative to the current. This angle allows him to effectively counteract the current while maintaining a straight path across the river.
Community Answer
A man wants to cross the river to an exactly opposite point on the oth...
45 degree
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A man wants to cross the river to an exactly opposite point on the other bank if he can row his boat with root two times the velocity of water current at what angle to the current must be keep the word moving?
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A man wants to cross the river to an exactly opposite point on the other bank if he can row his boat with root two times the velocity of water current at what angle to the current must be keep the word moving? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about A man wants to cross the river to an exactly opposite point on the other bank if he can row his boat with root two times the velocity of water current at what angle to the current must be keep the word moving? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A man wants to cross the river to an exactly opposite point on the other bank if he can row his boat with root two times the velocity of water current at what angle to the current must be keep the word moving?.
A man wants to cross the river to an exactly opposite point on the other bank if he can row his boat with root two times the velocity of water current at what angle to the current must be keep the word moving? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about A man wants to cross the river to an exactly opposite point on the other bank if he can row his boat with root two times the velocity of water current at what angle to the current must be keep the word moving? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A man wants to cross the river to an exactly opposite point on the other bank if he can row his boat with root two times the velocity of water current at what angle to the current must be keep the word moving?.
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