Multiple choice

A boat crosses a river from port A to port B, which are just on the opposite side. The speed of the water is ${ V }{ W }$ and that of boat is ${ V }{ B }$ relative to still water. Assume ${ V }{ B }$ = 2${ V }{ W }$. What is the time taken by the boat, if it has to cross the river directly on the AB line

  1. $\frac { 2D }{ { V }_{ B }\sqrt { 3 } } $
  2. $\frac { \sqrt { 3 } D }{ 2{ V }_{ B } } $
  3. $\frac { D }{ { { V }_{ B } }\sqrt { 2 } } $
  4. $\frac { D\sqrt { 2 } }{ { { V }_{ B } } } $
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A Correct answer
Explanation

To cross directly, the boat must head upstream at an angle. The velocity component perpendicular to the bank must be V_y = sqrt(V_B^2 - V_W^2). Since V_B = 2V_W, V_y = sqrt(4V_W^2 - V_W^2) = V_W * sqrt(3). Since V_W = V_B/2, V_y = (V_B/2) * sqrt(3). Time = D / V_y = 2D / (V_B * sqrt(3)).

AI explanation

To travel directly across the river on line AB, the boat must aim upstream at an angle so its upstream component cancels the river speed, requiring VB*cos(theta) = VW. Since VB = 2*VW, we have cos(theta) = 1/2, meaning theta is 60 degrees and the sine of the angle is sqrt(3)/2. The effective velocity across the river is the perpendicular component of the boat's velocity, calculated as VB*sin(60) = VB*sqrt(3)/2. The time taken to cross the distance D is the distance divided by this effective velocity, yielding 2D / (VB * sqrt(3)). The result is 2D / (VB * sqrt(3)).

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