A boat which can travel at 5m/s in still water is crossing a river 200 m wide.the rate of the flow of the river is 2 m/s assumed uniform at every point in the river. Point A and B are directly opposite to each other across the river.

a) If the boat leaves A and steer towards B, at what speed it will travel and at what point it will reach the opposite bank?
b) If the boat needs to travel towards B, in which direction it should be steered and the speed at which it will travel?

23degree

a) To find the speed at which the boat will travel and the point it will reach on the opposite bank, we need to consider the vector arrangement of the boat's motion.

Let's break down the boat's velocity into two components:
1. Velocity of the boat relative to still water: 5 m/s
2. Velocity of the river flow: 2 m/s

When the boat travels from Point A towards Point B, it needs to counteract the river's flow. The velocity of the river is perpendicular to the course of the boat. Therefore, to counteract it, the boat should steer at an angle relative to the river flow.

To calculate the diagonal velocity of the boat, we will use the Pythagorean theorem:
Diagonal velocity^2 = (5 m/s)^2 + (2 m/s)^2
Diagonal velocity^2 = 25 + 4
Diagonal velocity^2 = 29
Diagonal velocity ≈ √29 ≈ 5.39 m/s (rounded to two decimal places)

The boat will travel at a speed of approximately 5.39 m/s when it steers towards Point B.

To find the point at which the boat will reach the opposite bank, we need to consider the time it takes to cross the river. Since the boat's speed is the diagonal velocity, the time taken to cross the river is given by:
Time = Distance / Velocity
Time = 200 m / 5.39 m/s ≈ 37.09 s (rounded to two decimal places)

Since the boat steers directly towards Point B, it will reach the opposite bank after approximately 37.09 seconds.

b) If the boat needs to travel towards Point B (along the river flow), it should be steered at an angle parallel to the river flow. In this case, the boat needs to counteract only the perpendicular component of the river's flow. The boat's velocity relative to the still water remains the same, i.e., 5 m/s.

To find the resultant velocity of the boat in this case, we only consider the perpendicular component of the river flow, which is 2 m/s.

To calculate the resultant velocity, we subtract the river's flow velocity from the boat's velocity relative to still water:
Resultant velocity = Boat's velocity - River flow velocity
Resultant velocity = 5 m/s - 2 m/s = 3 m/s

Therefore, if the boat needs to travel towards Point B, it should be steered in the direction of the river flow with a speed of 3 m/s.