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?

a)by Phythagorous

c^2 =a^2 + b^2
= 5^2+ 2^2=25+4=29
c=5.38m/s
b)sin (theta)=2/5
theta = sin^-1 (2/5)
theta=23.57 degree

To answer these questions, we need to consider the relative motion of the boat with respect to the river.

a) When the boat is moving from Point A to Point B, it is traveling both forward due to its own speed and sideways due to the river's flow.

The component of the boat's velocity along the river's flow would be the velocity of the river itself, which is 2 m/s.

To find the resultant velocity of the boat, we can use vector addition. We form a right-angled triangle with the velocity of the boat in still water as the hypotenuse, the boat's velocity along the river's flow as one side, and the boat's resultant velocity as the other side.

Let's call the boat's resultant velocity V. Using the Pythagorean theorem, we can find V:
V^2 = (5 m/s)^2 + (2 m/s)^2
V^2 = 25 m^2/s^2 + 4 m^2/s^2
V^2 = 29 m^2/s^2
V ≈ √29 ≈ 5.39 m/s

Therefore, the boat will travel at approximately 5.39 m/s.

To find the point at which the boat will reach the opposite bank, we can use the formula:
time = distance / velocity

The distance across the river is given as 200 m. The boat's resultant velocity (V) represents its speed across the river. Therefore, the time taken to cross the river would be:
time = 200 m / 5.39 m/s ≈ 37.1 seconds

So, the boat will reach the opposite bank approximately 37.1 seconds after leaving Point A.

b) Now let's consider the scenario where the boat needs to travel towards Point B. Since the boat needs to counter the river's flow, it should be steered slightly upstream against the river's flow.

To determine the direction in which the boat should be steered, we can use the concept of the resultant velocity. The boat's resultant velocity should be directed towards the opposite bank, which is Point B.

The angle between the resultant velocity and the river's flow represents the direction in which the boat should be steered. Since the river's flow is directly sideways, the angle would be less than 90 degrees. By steering the boat slightly upstream, the boat's resultant velocity will have a component directed towards the opposite bank.

The magnitude of the boat's resultant velocity can be determined using the same method as in part a. That is, using vector addition. Substitute the boat's resultant velocity into the formula given earlier to determine its magnitude.