A butterfly is flying with velocity
10i +12jˆ m/s and wind is blowing along x axis with velocity u. If butterfly starts motion from A and after some time reaches
point B, find the value of u.
Answer this Question
To find the value of the wind velocity, u, we need to use the given information about the butterfly's velocity and the knowledge of relative motion.
Let's break down the problem step by step:
1. We are given the butterfly's velocity as 10i + 12j m/s. Here, i and j are the unit vectors along the x and y directions, respectively.
2. Since the wind is blowing along the x-axis, it will only affect the butterfly's velocity in the x-direction.
3. Let's assume the wind velocity u, in the same unit (m/s), acts only in the x-direction. Therefore, we can represent the wind vector as u + 0j.
4. Now, we can calculate the resulting velocity of the butterfly (v) by adding the wind vector to the butterfly's velocity:
v = (10 + u)i + 12j
Next, we need to find the time it takes for the butterfly to travel from point A to point B. Let's denote this time as 't' (in seconds).
5. The displacement of the butterfly can be calculated using the formula:
displacement = velocity * time
6. The displacement vector is given by the difference between the position vectors of B and A:
displacement = B - A
7. Since the butterfly starts at point A, its initial position vector is 0i + 0j.
8. Let's assume the final position vector of point B is bxi + byj. Therefore, the displacement vector is given by:
displacement = bxi + byj - (0i + 0j)
= bxi + byj
9. Now, using the formula mentioned before, we equate the displacement vector to the product of velocity and time:
bxi + byj = (10 + u)i + 12j * t
10. Comparing the i and j components, we get two equations:
bx = (10 + u)t
by = 12t
11. Since the butterfly's motion is in a straight line, the displacements in the x and y directions must be proportional to the time t.
Now, we have two equations and two unknowns (bx, by, and t) that need to be solved to find u.
12. Divide the two equations to eliminate t:
(bx/by) = ((10 + u)t) / (12t)
13. Cancel out the 't' terms:
(bx/by) = (10 + u)/12
14. Cross-multiply:
12 * (bx) = (10 + u) * (by)
15. Now, substitute the given value of bx and by:
12 * 10 = (10 + u) * 12
16. Simplify and solve for u:
120 = 120 + 12u
12u = 0
u = 0
Therefore, the value of the wind velocity, u, is 0 m/s.