Physics, 14.07.2019 15:00 neazisa

Your dog is running around the grass in your back yard. he undergoes successive displacements 3.20 m south, 8.34 m northeast, and 13.0 m west. what is the resultant displacement? (take the +x direction to be east.)

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Answer from: bettybales1986

Amirmkr_1995(at)yahoo.com

Displacement =7.6m
Your dog is running around the grass in your back yard. he undergoes successive displacements 3.20 m

Answer from: cesarcastellan9

The displacement of the dog is $\boxed{7.60\,{\text{m}}}$ and the direction of resultant displacement is $\boxed{20.72^\circ }$ north of west.

Further Explanation:

The dog moves across the lawn as shown in figure. The dog moves from the point $A \to B \to C \to D$ . The final displacement of the dog will be the distance .

As the dog moves in a straight line from A to B, the displacement of the dog in the direction will be $3.2\,{\text{m}}$ from its initial position. Now, the dog turns and moves along the north-east direction and covers a distance of $8.34\,{\text{m}}$ .

The displacement of dog from point B to C in the horizontal direction is:

$\begin{aligned}B{C_x}&=8.34\cos 45^\circ\\&= 5.89\,{\text{m}}\\\end{aligned}$

The displacement of dog from point B to C in the vertical direction is:

$\begin{aligned}B{C_y}&= 8.34\sin 45^\circ\\&= 5.89\,{\text{m}}\\\end{aligned}$

Now from point C, the dog moves horizontally towards the point D and covers a distance of $13\,{\text{m}}$ in the respective direction.

So the final displacement of the dog in the horizontal direction from its initial position is:

$\begin{aligned}{d_x} &= 13 - 5.89\\&= 7.11\,{\text{m}}\\\end{aligned}$

Displacement of the dog in the vertical direction from its initial position is:

$\begin{aligned}{d_y}&= 5.89-3.2\\&=2.69\,{\text{m}}\\\end{aligned}$

Thus, the magnitude of total displacement of the dog from initial position is:

$\begin{aligned}AD&=\sqrt{{{\left( {7.11}\right)}^2} + {{\left({2.69} \right)}^2}}\\&= \sqrt {57.78}\\&= 7.60\,{\text{m}}\\\end{aligned}$

The direction of displacement of the dog from its initial position is:

$\begin{aligned}\theta&= {\tan ^{ - 1}}\left({\frac{{{d_y}}}{{{d_x}}}}\right)\\&= {\tan ^{ - 1}}\left({\frac{{2.69}}{{7.11}}}\right)\\&= 20.72^\circ \,{\text{north}}\,{\text{of}}\,{\text{west}}\\\end{aligned}$

Thus, the displacement of the dog is $\boxed{7.60\,{\text{m}}}$ and the direction of resultant displacement is $\boxed{20.72^\circ }$ north of west.

Learn More:

1. A toy train rolls around a horizontal 1.0-m-diameter track. The coefficient of rolling friction is 0.10

2. Forces of attraction limit the motion of particles most in

3. Think about what subsequently happens to the ketchup, which is initially at rest, and use Newton’s first law

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.