An ice skater skates 13.3 m southwest, then 26.6 m east, and finally 39.9 m in a direction 51.5◦ north of east.

Find the magnitude of the displacement required to bring the skater back to her starting point.
Answer in units of m

103.8

123.4

To find the magnitude of the displacement required to bring the skater back to her starting point, we need to consider the combined effect of the individual displacements.

Let's break down the given displacements into their x and y components:

1. The first displacement of 13.3 m southwest is a combination of movements in the x and y directions. Since it's southwest, we can use trigonometry to find the x and y components. The angle between the southwest direction and the x-axis is 45 degrees.

The x-component can be found by multiplying the magnitude of the displacement by the cosine of the angle:
x₁ = 13.3 m * cos(45°) = 9.4 m (approximately)

The y-component can be found by multiplying the magnitude of the displacement by the sine of the angle:
y₁ = 13.3 m * sin(45°) = 9.4 m (approximately)

2. The second displacement of 26.6 m east is entirely in the positive x-direction, so the x-component (x₂) is 26.6 m, and the y-component (y₂) is 0 m.

3. The third displacement of 39.9 m in a direction 51.5° north of east can also be broken down into x and y components. Let's find those:

The x-component can be found by multiplying the magnitude of the displacement by the cosine of the angle:
x₃ = 39.9 m * cos(51.5°)

The y-component can be found by multiplying the magnitude of the displacement by the sine of the angle:
y₃ = 39.9 m * sin(51.5°)

Now, let's sum up the x and y components to find the net displacement:

Net x-component = x₁ + x₂ + x₃
Net y-component = y₁ + y₂ + y₃

Finally, the magnitude of the net displacement (D) can be calculated using the Pythagorean theorem:

D = √[(Net x-component)² + (Net y-component)²]

Plug in the values for x₁, y₁, x₂, y₂, x₃, y₃, and calculate the magnitude of the net displacement D.