a butterfly is flying with velocity 10i+12j m/s and wind is blowing along x axis with a velocity u.if butterfly starts motion from a and after sometime reaches time b.find the value of u.
Answers
Given:
Velocity of butterfly, v = (10 i + 12 j) m/s.
Velocity of wind, u = u i m/s
Relative velocity of butterfly V = v + u
V = [(10 +u ) i + 12 j ] m/s
Let the position vector of butterfly at starting point be a = (x₁ i +y₁ j) and the position vector of the butterfly at final point be b = (x₂ i +y₂ j)
Now, we shall calculate, net displacement vector ab = (x₂ -x₁ ) i + (y₂ - y₁ ) j -----(i)
Let 't' be the time in which, butterfly covers the displacement ab
Net displacement of butterfly ab = V× t
= [(10 +u ) i + 12 j ] m/s × (t s)
= [(10 +u ) i + 12 j ] × t m ---------(ii)
From equation (i) and equation (ii), we have,
[(10 +u ) i + 12 j ] × t = (x₂ -x₁ ) i + (y₂ - y₁ ) j
or, u = [(x₂ -x₁ ) / t] i + [(y₂ - y₁ ) / t] j - (10 i + 12 j)
or, u = { [(x₂ -x₁ ) / t] - 10} i + { [(y₂ - y₁ ) / t] - 12 } j
Answer:
Given data states that the velocity of butterfly be v = (10 i + 12 j) m/s, and the wind is blowing along x-axis therefore velocity of wind be u = u i m/s.
We can say from this that relative velocity of butterfly be V = v + u and is equal to,
V = [(10 + u ) i + 12 j ] m/s.
In addition given butterfly starts from point a and reaches b therefore we can say the position vector for a and b be a = (x₁ i +y₁ j) and b = (x₂ i +y₂ j) respectively.
Thereby, net displacement vector be ab = (x₂ -x₁ ) i + (y₂ - y₁ ) j.
The time taken to reach a to b be time t therefore, net displacement of butterfly, ab = V× t
=> ab = [(10 +u ) i + 12 j ] × (t)
=> ab = [(10 +u ) i + 12 j ] × t.
Solving the earlier net displacement equation with the latter, we get
- [(10 +u ) i + 12 j ] × t = (x₂ -x₁ ) i + (y₂ - y₁ ) j
=> u = [(x₂ -x₁ ) / t] i + [(y₂ - y₁ ) / t] j - (10 i + 12 j)
=> u = { [(x₂ -x₁ ) / t] - 10} i + { [(y₂ - y₁ ) / t] - 12 } j