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A man desires to swim across the river in shortest time. The velocity of river water is 3 km h¹. He can swim in still water at 6 km h-¹. At what angle with the velocity of flow of the river?
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A man desires to swim across the river in shortest time. The velocity ...
To determine the angle at which the man should swim to cross the river in the shortest time, we must analyze the components of his swimming velocity and the river's current.
Understanding the Velocities
- The velocity of the river (Vr) = 3 km/h
- The man's swimming speed in still water (Vs) = 6 km/h
Components of Velocity
- When the man swims at an angle (θ) with respect to the current, his swimming velocity can be broken into two components:
- Upstream component: Vs * cos(θ)
- Across the river component: Vs * sin(θ)
Net Velocity Across the River
- To cross directly, the upstream component must counter the river's current:
Vs * cos(θ) = Vr
- This implies:
cos(θ) = Vr / Vs = 3 / 6 = 0.5
- Therefore, θ = cos⁻¹(0.5) = 60°.
Calculating the Required Angles
- The angle θ indicates the direction to swim relative to the current to maintain a straight path across the river. The swimmer will need to swim upstream at an angle of 60° to offset the river's current while still making forward progress across the river.
Conclusion
- The optimal angle for the man to swim to cross the river in the shortest time, considering the river's flow, is 60° upstream from the direct line across the river. This angle maximizes his effective velocity across the river while compensating for the current.
Understanding the Velocities
- The velocity of the river (Vr) = 3 km/h
- The man's swimming speed in still water (Vs) = 6 km/h
Components of Velocity
- When the man swims at an angle (θ) with respect to the current, his swimming velocity can be broken into two components:
- Upstream component: Vs * cos(θ)
- Across the river component: Vs * sin(θ)
Net Velocity Across the River
- To cross directly, the upstream component must counter the river's current:
Vs * cos(θ) = Vr
- This implies:
cos(θ) = Vr / Vs = 3 / 6 = 0.5
- Therefore, θ = cos⁻¹(0.5) = 60°.
Calculating the Required Angles
- The angle θ indicates the direction to swim relative to the current to maintain a straight path across the river. The swimmer will need to swim upstream at an angle of 60° to offset the river's current while still making forward progress across the river.
Conclusion
- The optimal angle for the man to swim to cross the river in the shortest time, considering the river's flow, is 60° upstream from the direct line across the river. This angle maximizes his effective velocity across the river while compensating for the current.
Community Answer
A man desires to swim across the river in shortest time. The velocity ...
But answer is given is 90degree
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A man desires to swim across the river in shortest time. The velocity of river water is 3 km h¹. He can swim in still water at 6 km h-¹. At what angle with the velocity of flow of the river?
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A man desires to swim across the river in shortest time. The velocity of river water is 3 km h¹. He can swim in still water at 6 km h-¹. At what angle with the velocity of flow of the river? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A man desires to swim across the river in shortest time. The velocity of river water is 3 km h¹. He can swim in still water at 6 km h-¹. At what angle with the velocity of flow of the river? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A man desires to swim across the river in shortest time. The velocity of river water is 3 km h¹. He can swim in still water at 6 km h-¹. At what angle with the velocity of flow of the river?.
A man desires to swim across the river in shortest time. The velocity of river water is 3 km h¹. He can swim in still water at 6 km h-¹. At what angle with the velocity of flow of the river? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A man desires to swim across the river in shortest time. The velocity of river water is 3 km h¹. He can swim in still water at 6 km h-¹. At what angle with the velocity of flow of the river? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A man desires to swim across the river in shortest time. The velocity of river water is 3 km h¹. He can swim in still water at 6 km h-¹. At what angle with the velocity of flow of the river?.
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