two swimmwrs leave point A on one bank of the river to reach poit B on the other bank.onne of them crossers the river along the the straight line AB while the other swims at right angles to the river and then walks the distance that he has been carried away by the stream to get to point B.find the velocity with which the man should walk if both swimmers reach their destination simultaneously.the stream velocity is 2km/hr and velocity of each swimmer wrt water is 2.5km/hr

To solve this problem, we need to analyze the movements of both swimmers and how they interact with the river's current. Let's break it down step by step.

Understanding the Scenario

We have two swimmers starting from point A on one bank of the river and aiming to reach point B on the opposite bank. The first swimmer swims directly across the river along the straight line AB, while the second swimmer swims perpendicular to the river's flow and then walks to reach point B. The key here is that both swimmers arrive at point B at the same time.

Given Information

• Stream velocity (Vs) = 2 km/hr
• Velocity of each swimmer relative to water (Vw) = 2.5 km/hr

Calculating the Time Taken by Each Swimmer

Let's denote the width of the river as 'd'. The first swimmer, who swims directly across, will take a certain time to reach the other side. The time taken (T1) can be calculated using the formula:

T1 = d / Vw

Since the swimmer's velocity is 2.5 km/hr, we can express this as:

T1 = d / 2.5

Movement of the First Swimmer

While swimming straight across, the first swimmer is also being carried downstream by the current. The distance he is carried downstream (D1) can be calculated as:

D1 = Vs * T1 = 2 * (d / 2.5) = (2d) / 2.5 = 0.8d

Analyzing the Second Swimmer's Path

The second swimmer swims perpendicular to the current. He will swim a distance 'd' across the river, which takes the same time T2 as the first swimmer:

T2 = d / Vw = d / 2.5

After reaching the opposite bank, he must walk the distance he was carried downstream, which is D1 = 0.8d. The time taken to walk this distance (Tw) can be expressed as:

Tw = D1 / Vwalk

Setting Up the Equation

Since both swimmers reach point B simultaneously, we can set T1 equal to the total time taken by the second swimmer:

T2 + Tw = T1

d / 2.5 + 0.8d / Vwalk = d / 2.5

Solving for Walking Velocity

To find the walking velocity (Vwalk), we can rearrange the equation:

0.8d / Vwalk = 0

This simplifies to:

0.8d / Vwalk = 0

Now, isolating Vwalk gives us:

Vwalk = 0.8d / (d / 2.5)

Vwalk = 0.8 * 2.5 = 2 km/hr

Final Result

Thus, the velocity with which the second swimmer should walk to reach point B simultaneously with the first swimmer is 2 km/hr.

This problem illustrates the interplay between swimming and walking in the context of a current, highlighting how different paths can lead to the same destination when timed correctly.