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A boat moves relative to water with a velocity which is N= 2.0 times less then the river flow velocity. At what angle to the stream direction must the boat move to minimize the drifting?

A boat moves relative to water with a velocity which is N= 2.0 times less then the river flow velocity. At what angle to the stream direction must the boat move to minimize the drifting?

Grade:12

1 Answers

SOURAV MISHRA
37 Points
8 years ago

in this situation the drift cannot be zero because the speed of the boat realtive to water is less than the stream speed.

so in order to minimise drifting the angle with which the boat must move with the stream direction is 120°.

i will explain the result.

let the width of the river be d

speed of stream be v and the speed of the boat realtive to water be u.

and the angle with the vertical at which the boat must move for minimum drifting is θ. here vertical means the direction perpendicular to the stream flow that is straight across the river along which the drifting is zero.

the component of the boat velocity which is along the stream that is which causes the drifting is v - usinθ.

the component of the velocity of the boat which helps to cross the river is ucosθ.

note that here we are talking about the velocity of the boat as seen from the ground.

the time taken to cross the river is the time in which the boat has drifted along the river. this is equal to d/ucosθ.

so the drift of the boat is (v - usinθ)*(d/ucosθ). 

differentiating this with respect to time and equating to zero we get the angle θ for minimum drifting as sin-1(u/v).

so the angle with the direction of the stream is 90º + sin-1(u/v).

in this case we have u = v/2 so the angle is 120º.

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