The magnitude of resultant displacement of the dog is \(\boxed{9.91\,{\text{m}}}\) and the direction of resultant displacement is \(\boxed{13.89^\circ }\) north of west.
Further Explanation:
The displacement of the dog is as shown in the figure attached below. The dog moves from the point \(A\to B\to C\to D\) .
As the dog moves from A to B, it is a straight line and it is displaced by a distance of \(3.6\,{\text{m}}\) from its initial position. After that, the dog moves by a distance of \(8.46\,{\text{m}}\) in the north-east direction.
The resultant displacement of the dog from point B to point C in the horizontal direction is:
\(\begin{aligned}B{C_x}&=8.46\cos45^\circ\\&=5.98\,{\text{m}}\\\end{aligned}\)
The displacement from B to C in vertical direction is:
\(\begin{aligned}B{C_y}&=8.46\sin45^\circ\\&=5.98\,{\text{m}}\\\end{aligned}\)
Then after this displacement, the dog moves from point C to point D in the horizontal direction.
So the final displacement of the dog in the horizontal direction is:
\(\begin{aligned}{d_x}&=15.6-5.98\\&=9.62\,{\text{m}}\\\end{aligned}\)
Displacement of the dog in the vertical direction is:
\(\begin{aligned}{d_y}&=5.98-3.6\\&=2.38\,{\text{m}}\\\end{aligned}\)
Thus, the magnitude of total displacement of the dog from initial position is:
\(\begin{aligned}AD&=\sqrt{{{\left({9.62}\right)}^2}+{{\left({2.38}\right)}^2}}\\&=\sqrt{98.21}\\&=9.91\,{\text{m}}\\\end{aligned}\)
The direction of final displacement of the dog is:
\(\begin{aligned}\theta&={\tan ^{-1}}\left({\frac{{{d_y}}}{{{d_x}}}}\right)\\&={\tan^{-1}}\left({\frac{{2.38}}{{9.62}}}\right)\\&=13.89^\circ\,{\text{north}}\,{\text{of}}\,{\text{west}}\\\end{aligned}\)
Thus, the magnitude of resultant displacement of the dog is \(\boxed{9.91\,{\text{m}}}\) and the direction of resultant displacement is \(\boxed{13.89^\circ }\) north of west.
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Answer Details:
Grade: High School
Subject: Physics
Chapter: Distance and Displacement
Keywords:
Dog is running, grass, in your backyard, successive displacements, 3.60m south, 8.46m north east, 15.6 m west, resultant displacement, horizontal direction, vertical direction.
