Problem Statement: A butterfly is flying with velocity of 10i^ 12j^ and wind is blowing along X axis with velocity u. The butterfly starts moving and after sometime reaches a point. Find the value of u.

Solution:

Step 1: Understand the problem and identify the given parameters

The problem is about a butterfly flying with a velocity of 10i^ 12j^ and wind blowing along the X-axis with a velocity u. The butterfly starts moving and after some time, it reaches a point.

Given parameters:
- Velocity of butterfly = 10i^ 12j^
- Velocity of wind = u

Step 2: Determine the net velocity of the butterfly

The net velocity of the butterfly can be determined by adding the velocity of the butterfly and the velocity of the wind.

Net velocity of butterfly = Velocity of butterfly + Velocity of wind

Since the velocity of the butterfly is given as 10i^ 12j^ and the velocity of the wind is along the X-axis (i^), we can rewrite the above equation.

Net velocity of butterfly = 10i^ + (u * 0j^)

Net velocity of butterfly = 10i^ + 0j^ + u * 0j^

Net velocity of butterfly = 10i^ + u * 0j^

Net velocity of butterfly = 10i^

Step 3: Determine the time taken by the butterfly to reach the point

Let's assume that the butterfly starts at the origin (0, 0) and reaches the point (x, y) after time t.

Using the formula for displacement, we can write:

Displacement = Net velocity of butterfly * time taken

(x - 0)i^ + (y - 0)j^ = 10i^ * t

x = 10t
y = 0

Step 4: Determine the value of u

Now, we know that the butterfly reaches the point (x, y) after time t. From step 3, we know that x = 10t and y = 0.

Since the velocity of the wind is along the X-axis (i^), the net displacement of the butterfly along the X-axis must be equal to x.

Net displacement along X-axis = x = 10t

Using the formula for displacement again, we can write:

Displacement along X-axis = Velocity of butterfly along X-axis + Velocity of wind along X-axis * time taken

10t = 10 + u * t

Simplifying the above equation, we get:

10t - u * t = 10

t(10 - u) = 10

t = 10 / (10 - u)

From step 3, we know that x = 10t. Substituting the value of t in this equation, we get:

x = 10 / (10 - u)

10 / (10 - u) = 10

(10 - u) = 1

u = 9

Step 5: Conclusion

Therefore, the value of u is 9. The wind is blowing along the X-axis with a velocity of 9.

6m/s