A swimmer wants to cross a river from point A | vedclass

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Question

(A) Let the velocity of the river be $\vec{v}_r$ and the velocity of the swimmer with respect to the river be $\vec{v}_{sr}$.Given that the magnitudes

Vedclass · Accepted Answer

$30$

Vedclass · Answer

(A) Let the velocity of the river be $\vec{v}_r$ and the velocity of the swimmer with respect to the river be $\vec{v}_{sr}$.Given that the magnitudes are equal,let $|\vec{v}_r| = |\vec{v}_{sr}| = v$.The resultant velocity $\vec{v}_s = \vec{v}_{sr} + \vec{v}_r$ must be directed along the line $AB$ to reach point $B$.Since the magnitudes of the two vectors $\vec{v}_{sr}$ and $\vec{v}_r$ are equal,their resultant vector $\vec{v}_s$ bisects the angle between them.Let the angle between the river flow (horizontal) and the line $AB$ be $\alpha = 30^{\circ}$.Let the angle between the swimmer's velocity $\vec{v}_{sr}$ and the line $AB$ be $	heta$.Then the angle between the swimmer's velocity $\vec{v}_{sr}$ and the river flow $\vec{v}_r$ is $(	heta + 30^{\circ})$.Since the resultant $\vec{v}_s$ (which is along $AB$) bisects this angle,the angle between $\vec{v}_s$ and $\vec{v}_r$ must be equal to the angle between $\vec{v}_s$ and $\vec{v}_{sr}$.Therefore,$30^{\circ} = 	heta$.Thus,the required angle $	heta$ is $30^{\circ}$.

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