The displacement of the dog is [tex]\boxed{7.60\,{\text{m}}}[/tex] and the direction of resultant displacement is [tex]\boxed{20.72^\circ }[/tex] north of west.
Further Explanation:
The dog moves across the lawn as shown in figure. The dog moves from the point [tex]A \to B \to C \to D[/tex]. The final displacement of the dog will be the distance [tex]AD[/tex].
As the dog moves in a straight line from A to B, the displacement of the dog in the direction will be [tex]3.2\,{\text{m}}[/tex] from its initial position. Now, the dog turns and moves along the north-east direction and covers a distance of [tex]8.34\,{\text{m}}[/tex].
The displacement of dog from point B to C in the horizontal direction is:
[tex]\begin{aligned}B{C_x}&=8.34\cos 45^\circ\\&= 5.89\,{\text{m}}\\\end{aligned}[/tex] Â
The displacement of dog from point B to C in the vertical direction is:
[tex]\begin{aligned}B{C_y}&= 8.34\sin 45^\circ\\&= 5.89\,{\text{m}}\\\end{aligned}[/tex] Â
Now from point C, the dog moves horizontally towards the point D and covers a distance of [tex]13\,{\text{m}}[/tex] in the respective direction.
So the final displacement of the dog in the horizontal direction from its initial position is:
 [tex]\begin{aligned}{d_x} &= 13 - 5.89\\&= 7.11\,{\text{m}}\\\end{aligned}[/tex]
Displacement of the dog in the vertical direction from its initial position is:
[tex]\begin{aligned}{d_y}&= 5.89-3.2\\&=2.69\,{\text{m}}\\\end{aligned}[/tex] Â
Thus, the magnitude of total displacement of the dog from initial position is:
[tex]\begin{aligned}AD&=\sqrt{{{\left( {7.11}\right)}^2} + {{\left({2.69} \right)}^2}}\\&= \sqrt {57.78}\\&= 7.60\,{\text{m}}\\\end{aligned}[/tex] Â
The direction of displacement of the dog from its initial position is:
[tex]\begin{aligned}\theta&= {\tan ^{ - 1}}\left({\frac{{{d_y}}}{{{d_x}}}}\right)\\&= {\tan ^{ - 1}}\left({\frac{{2.69}}{{7.11}}}\right)\\&= 20.72^\circ \,{\text{north}}\,{\text{of}}\,{\text{west}}\\\end{aligned}[/tex] Â
Thus, the displacement of the dog is [tex]\boxed{7.60\,{\text{m}}}[/tex] and the direction of resultant displacement is [tex]\boxed{20.72^\circ }[/tex] north of west.
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Answer Details:
Grade: High School
Subject: Physics
Chapter: Distance and Displacement
Keywords: Â 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.
