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A man wishes to cross a river to an exac...

A man wishes to cross a river to an exactly opposite point on the other bank. if the can swim with twice the velocity of the current, the inclination to the current of the direction he should swim is

A

`30^(@)`

B

`45^(@)`

C

`60^(@)`

D

`120^(@)`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem of a man wishing to cross a river to an exactly opposite point on the other bank while swimming with a velocity that is twice that of the current, we can follow these steps: ### Step 1: Define Variables Let: - \( v_s \) = velocity of the swimmer - \( v_c \) = velocity of the current - The swimmer's velocity is given as \( v_s = 2v_c \). ### Step 2: Set Up the Problem The swimmer needs to swim at an angle \( \theta \) upstream to counteract the downstream current. The goal is to ensure that the resultant velocity of the swimmer has no downstream component, allowing him to reach the point directly across the river. ### Step 3: Resolve Velocities The swimmer's velocity can be resolved into two components: 1. A component perpendicular to the current (across the river): \( v_s \cos(\theta) \) 2. A component parallel to the current (against the current): \( v_s \sin(\theta) \) ### Step 4: Equate the Components To cross directly to the opposite point, the upstream component of the swimmer's velocity must equal the downstream velocity of the current: \[ v_s \sin(\theta) = v_c \] ### Step 5: Substitute the Swimmer's Velocity Substituting \( v_s = 2v_c \) into the equation: \[ 2v_c \sin(\theta) = v_c \] ### Step 6: Simplify the Equation Dividing both sides by \( v_c \) (assuming \( v_c \neq 0 \)): \[ 2 \sin(\theta) = 1 \] ### Step 7: Solve for \( \sin(\theta) \) \[ \sin(\theta) = \frac{1}{2} \] ### Step 8: Find the Angle \( \theta \) To find \( \theta \), we take the inverse sine: \[ \theta = \sin^{-1}\left(\frac{1}{2}\right) = 30^\circ \] ### Conclusion The inclination to the current of the direction he should swim is \( 30^\circ \). ---
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