A man can swim at a speed of 3km/h in still water. He wants to cross a 500m wide river flowing at a speed of 2km/h. He keeps himself always at an angle of 120 degree with the river flow while swimming. Find the time he takes to cross the river and at what point on the opposite bank will he arrive.

Repliers in under 10 minutes with complete diagram and steps are rated as Genius, those under 20 minutes are Stupendous, afterall every correct replier will be reated as good at kinematics by me. NOTE : I could solve it with all the mentioned conditions in 7 minutes 29 seconds. Lets see how many of you are quicker than me. Finally : IT IS NOT TOUGH SO DO GIVE IT A TRY. I'm waiting.

To solve this problem, we need to analyze the swimmer's motion in relation to the river's current. The swimmer's speed in still water is 3 km/h, and the river flows at 2 km/h. The swimmer maintains an angle of 120 degrees with respect to the direction of the river flow. Let's break this down step by step.

Understanding the Components of Motion

The swimmer's velocity can be broken down into two components: one that acts perpendicular to the river flow (across the river) and another that acts parallel to the river flow (downstream). To find these components, we can use trigonometric functions.

Calculating the Components of Velocity

Given that the swimmer swims at an angle of 120 degrees to the river flow, we can determine the components of his swimming velocity:

• Velocity across the river (V_y): This is the component that helps him cross the river. It can be calculated using the sine function:
• Velocity downstream (V_x): This is the component that moves him downstream with the current. It can be calculated using the cosine function:

Using the angle of 120 degrees:

• V_y = 3 km/h * sin(120°) = 3 km/h * (√3/2) = 2.598 km/h
• V_x = 3 km/h * cos(120°) = 3 km/h * (-1/2) = -1.5 km/h

Effective Velocity Across the River

The effective velocity across the river is simply V_y, which is approximately 2.598 km/h. The river's current adds a downstream component of 2 km/h to the swimmer's downstream motion.

Finding the Time to Cross the River

The width of the river is 500 meters, which we need to convert to kilometers for consistency in units:

• 500 m = 0.5 km

Now, we can calculate the time taken to cross the river using the formula:

Time = Distance / Velocity

Substituting the values:

Time = 0.5 km / 2.598 km/h ≈ 0.192 hours

To convert this into minutes:

0.192 hours * 60 minutes/hour ≈ 11.54 minutes

Determining the Downstream Displacement

While the swimmer is crossing the river, he is also being carried downstream by the river's current. We need to calculate how far downstream he travels during the time it takes to cross.

Using the downstream velocity:

Distance downstream = Velocity downstream * Time

Distance downstream = 2 km/h * 0.192 hours ≈ 0.384 km = 384 meters

Final Position on the Opposite Bank

When the swimmer reaches the opposite bank, he will have moved 384 meters downstream from his starting point. Therefore, the swimmer will arrive at a point 384 meters downstream from where he initially entered the water.

Summary of Results

In summary, the swimmer takes approximately 11.54 minutes to cross the river and will end up 384 meters downstream on the opposite bank. This problem illustrates the importance of vector components in analyzing motion in different directions, especially in the presence of currents or other forces.