A dog searching for a bone walks 3.50 m south, then 8.20 m north of east, and finally 15.0 m west.

(a) What is the magnitude of the dog's total displacement?
(b) What is the direction of the dog's total displacement where directly east is taken as
zero degrees and counter-clockwise is positive?

The answer for a is 7.9

To find the magnitude of the dog's total displacement, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let's break down the dog's displacements into components:

1. The dog walks 3.50 m south, which means it has a displacement of -3.50 m in the y-direction.
2. The dog walks 8.20 m north of east, which means it has a displacement of +8.20 m in the x-direction.
3. The dog walks 15.0 m west, which means it has a displacement of -15.0 m in the x-direction.

(a) To find the magnitude of the total displacement, we can calculate the net displacement in the x and y directions separately and then use the Pythagorean theorem to find the total displacement.

Net displacement in the x-direction = (+8.20 m) + (-15.0 m) = -6.80 m
Net displacement in the y-direction = -3.50 m

Total displacement = √((-6.80 m)² + (-3.50 m)²)
= √(46.24 m² + 12.25 m²)
= √58.49 m²
= 7.64921 m

Therefore, the magnitude of the dog's total displacement is approximately 7.65 m.

(b) To find the direction of the dog's total displacement, we can use trigonometry.

θ = tan⁻¹(net displacement in the y-direction/net displacement in the x-direction)

θ = tan⁻¹(-3.50 m/-6.80 m)
= tan⁻¹(0.5147)
= 28.540° (rounded to three decimal places)

Since directly east is taken as zero degrees and counter-clockwise is positive, the direction of the dog's total displacement is 28.540° counter-clockwise from east.

To find the magnitude of the dog's total displacement, we can use the concept of vector addition. We need to add the individual displacements in each direction.

(a) Magnitude of the dog's total displacement:
1. Start by drawing a diagram with labeled distances and directions. Place the starting point at the origin (0, 0).
2. The dog walks 3.50 m south, so draw an arrow downwards representing a displacement of 3.50 m.
3. The dog then walks 8.20 m north of east. Since north is directed upward and east to the right, the displacement will be a diagonal line in the northeast direction. Use the Pythagorean theorem to find the length of this displacement.
The vertical component of the displacement is 8.20 m * sin(45°) = 5.80 m upwards.
The horizontal component of the displacement is 8.20 m * cos(45°) = 5.80 m towards the right.
Draw an arrow connecting the end of the first displacement to the end point of the second displacement (5.80 m towards the right and 5.80 m upwards).
4. Lastly, the dog walks 15.0 m west. Draw an arrow towards the left with a length of 15.0 m.

Now, we have a triangle with three sides: 3.50 m downwards, 5.80 m upwards, and 15.0 m towards the left. To find the magnitude of the total displacement, we need to find the length of the resultant displacement vector.

Using the Pythagorean theorem:
Resultant displacement = sqrt((3.50 m - 15.0 m)^2 + (5.80 m)^2)

(b) To determine the direction of the dog's total displacement, we can use trigonometry. The direction is usually measured from the positive x-axis, where directly east is taken as zero degrees, and counter-clockwise is positive.

1. Calculate the angle between the resultant displacement and the positive x-axis:
Angle = arctan(5.80 m / (15.0 m - 3.50 m))

Now, you can use a calculator or a trigonometric table to find the angle in degrees.

Note: Ensure to convert the angle from radians to degrees if you are using a calculator that provides results in radians.

The answer to part (a) is the magnitude of the total displacement, and the answer to part (b) is the direction of the displacement in degrees.

8.2 at NE = 5.8N and 5.8E

(0,-3.5)+(5.8,5.8)+(-15,0) = (-9.2,2.3)

(a) sqrt(9.2^2 + 2.3^2) = 9.48
(b) arctan(2.3/-9.2) = 166 deg