A boat can travel at a speed of 3 m/s on still
water. A boatman wants to cross a river whilst covering the shortest possible distance. In what direction
should he row with respect to the bank if the speed
of the water is
a) 2 m/s,
b) 4 m/s?
Assume that the speed of the water is the same
everywhere
To determine the direction in which the boatman should row with respect to the bank, we need to analyze the velocity vectors of the boat in different scenarios.
Let's consider the boat's velocity vector relative to the water, which is 3 m/s in the direction of the boat's movement on still water. Since we are assuming the speed of the water is the same everywhere, we only need to consider the horizontal component of the boat's velocity.
a) When the speed of the water is 2 m/s:
If the boat rows directly across the river, perpendicular to the current, the water current will push the boat horizontally. The resulting velocity vector will be the vector sum of the boat's velocity relative to the water and the water's velocity. Using vector addition, we can find the resulting velocity:
Vresultant = Vboat + Vwater
Given that the boat's velocity relative to the water (Vboat) is 3 m/s and the water's velocity (Vwater) is 2 m/s, we can calculate the resulting velocity vector (Vresultant). The magnitude of Vresultant will be sqrt(3^2 + 2^2) = sqrt(13) m/s.
The boatman should row in a direction such that the boat's velocity vector is perpendicular to the river flow. This means the magnitude of the horizontal component of the boat's velocity should match the water's velocity of 2 m/s. Therefore, the boatman should row at an angle such that the horizontal component of the boat's velocity is 2 m/s.
b) When the speed of the water is 4 m/s:
Using the same principle as above, the boatman should row in a direction such that the horizontal component of the boat's velocity matches the water's velocity of 4 m/s. Therefore, the boatman should row at an angle such that the horizontal component of the boat's velocity is 4 m/s.
In both scenarios, the boatman should row at an angle with respect to the bank that allows the horizontal component of the boat's velocity to counteract the horizontal component of the water's velocity, resulting in the shortest possible distance crossed.