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Step-by-Step Detailed Solution
Step 1: Represent the velocities
A butterfly is flying northeast with speed $4\sqrt{2}\,\text{m/s}$.
Northeast indicates a $45^\circ$ angle with respect to both the north (y) and east (x) directions.
Meanwhile, wind is blowing at $1\,\text{m/s}$ from north to south (which is along negative y-axis).
Step 2: Resolve the butterfly’s velocity into components
The velocity of the butterfly in the northeast direction can be broken into x and y components:
$v_{x} = 4\sqrt{2} \cos 45^\circ$ and
$v_{y} = 4\sqrt{2} \sin 45^\circ$.
Since $\cos 45^\circ = \sin 45^\circ = \frac{1}{\sqrt{2}}$,
$v_{x} = 4\sqrt{2} \times \frac{1}{\sqrt{2}} = 4,$
$v_{y} = 4\sqrt{2} \times \frac{1}{\sqrt{2}} = 4.$
Thus, the butterfly's velocity components are $4\,\hat{i}$ and $4\,\hat{j}$.
Step 3: Determine the wind velocity
The wind is blowing at $1\,\text{m/s}$ from north to south. This implies its velocity vector is
$-1\,\hat{j}$.
Step 4: Calculate the resultant (net) velocity
The net velocity of the butterfly is the vector sum of the butterfly’s velocity and the wind’s velocity:
$v_{\text{net}} = (4\,\hat{i} + 4\,\hat{j}) + (0\,\hat{i} - 1\,\hat{j}) = 4\,\hat{i} + 3\,\hat{j}.$
Step 5: Compute the displacement in 3 seconds
Displacement $D$ is the product of net velocity and time:
$D = v_{\text{net}} \times t = (4\,\hat{i} + 3\,\hat{j}) \times 3 = 12\,\hat{i} + 9\,\hat{j}.$
Step 6: Find the magnitude of the displacement
$|D| = \sqrt{(12)^2 + (9)^2} = \sqrt{144 + 81} = \sqrt{225} = 15\,\text{m}.$
Therefore, the resultant displacement of the butterfly in 3 seconds is $15\,\text{m}.$
