What is the magnitude of displacement for a person that walks 6.0-km North-East, 4.0-km East, and 6.0-km South?
To find the magnitude of displacement, we first need to calculate the sum of the individual displacements in both the x and y directions.
Given:
1. The person walked 6.0 km North-East.
2. The person walked 4.0 km East.
3. The person walked 6.0 km South.
Let's break down each displacement into its component vectors:
1. The person walked 6.0 km North-East, which can be split into two components:
- North component: 6.0 km * sin(45°)
- East component: 6.0 km * cos(45°)
2. The person walked 4.0 km East, which has no vertical component (since it is purely horizontal).
3. The person walked 6.0 km South, which can be split into two components:
- South component: -6.0 km (-ve direction)
- East component: 0 km (since it is purely vertical)
Next, we add up the individual components:
Horizontal component = (6.0 km * cos(45°)) + (4.0 km) + (0 km) = (6.0√2 + 4.0) km
Vertical component = (6.0 km * sin(45°)) + (0 km) + (-6.0 km) = (6.0√2 - 6.0) km
Finally, we use the Pythagorean theorem to calculate the magnitude of displacement (D):
D = √(horizontal component^2 + vertical component^2)
D = √((6.0√2 + 4.0)^2 + (6.0√2 - 6.0)^2)
Note: √2 is approximately 1.4142.
Calculating this equation will provide us with the magnitude of displacement for the given scenario.