to a man walking at the rate of 3 kmph ,the rain appears to fall vertically .when he increases his speed to 6 kmph , it appears to meet him at an angle of 45 degreewith vertical . Find the speed of rain

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To solve the problem of the rain's speed based on the observations of the man walking at different speeds, we can use some basic principles of relative motion. Let's break this down step by step.

Understanding the Scenario

We have a man walking at two different speeds: 3 km/h and 6 km/h. At 3 km/h, the rain appears to fall vertically, which means that the horizontal component of the rain's velocity is equal to the man's speed. When he increases his speed to 6 km/h, the rain appears to meet him at a 45-degree angle with the vertical. This gives us two key pieces of information to work with.

Setting Up the Problem

Let's denote:

• V_r: the speed of the rain (in km/h)
• V_m1: the man's speed at the first observation (3 km/h)
• V_m2: the man's speed at the second observation (6 km/h)

From the first observation, when the man walks at 3 km/h, the rain appears to fall vertically. This means that the horizontal component of the rain's speed must equal the man's speed:

V_r_horizontal = V_m1 = 3 km/h

Analyzing the Second Observation

In the second scenario, when the man walks at 6 km/h, the rain appears to meet him at a 45-degree angle. This means that the horizontal and vertical components of the rain's speed are equal at this point. We can express this mathematically:

V_r_horizontal = V_m2 = 6 km/h

Since the rain appears to meet him at a 45-degree angle, we can also say:

V_r_vertical = V_r_horizontal

Finding the Speed of Rain

From the first observation, we have established that:

V_r_horizontal = 3 km/h

From the second observation, we know:

V_r_horizontal = 6 km/h

However, we need to reconcile these two observations. The key is to realize that the horizontal component of the rain's speed does not change; it remains constant at 3 km/h. Therefore, when the man walks faster, the vertical component of the rain's speed must adjust to maintain the 45-degree angle.

Using the relationship from the second observation, we can set up the following equations:

• At 45 degrees, we have: V_r_vertical = V_r_horizontal
• Thus, V_r_vertical = 6 km/h - 3 km/h = 3 km/h

Calculating the Result

Now we can find the total speed of the rain using the Pythagorean theorem:

V_r = √(V_r_horizontal² + V_r_vertical²)

Substituting the values we have:

V_r = √(3² + 3²) = √(9 + 9) = √18 = 3√2 km/h

Calculating this gives us approximately:

V_r ≈ 4.24 km/h

Final Thoughts

The speed of the rain is approximately 4.24 km/h. This problem illustrates how relative motion can be analyzed using basic principles of geometry and trigonometry, allowing us to deduce the speed of an object based on observations from a moving reference point.