Question

You walk south for 12.5 km, west for 8.9 km, and then north for 5 km. What is the magnitude and direction of your resultant displacement?

A) 15.6 km south
B) 9.4 km west
C) 7.5 km south
D) 3.6 km east

Answer 1

The magnitude of the resultant displacement is 15.6 km south.

Step-by-step explanation:

To find the magnitude and direction of the resultant displacement, we can use vector addition. We have three displacements: south for 12.5 km, west for 8.9 km, and north for 5 km.

First, let's consider the south and north displacements. Since they are in opposite directions, we can subtract the magnitude of the north displacement from the south displacement. 12.5 km - 5 km = 7.5 km south.

Next, let's consider the west displacement. We can add this west displacement to the resultant of the south and north displacements. 7.5 km south + 8.9 km west = 16.4 km in a diagonal direction.

To find the magnitude of the resultant displacement, we can use the Pythagorean theorem.

The magnitude is the square root of the sum of the squares of the components. √((7.5 km)² + (8.9 km)²) ≈ 11.9 km.

Lastly, to determine the direction of the resultant displacement, we can use trigonometry.

The direction can be found by taking the inverse tangent of the ratio of the north/south displacement and the west/south displacement.

arctan(8.9 km/7.5 km) ≈ 49.5 degrees west of south.

Therefore, the magnitude of the resultant displacement is approximately 11.9 km, and the direction is approximately 49.5 degrees west of south. None of the given options match the calculated values.

Answer 2

The resultant displacement after walking south, west, and then north is approximately 11.6 km in the direction 40 degrees south of west. The answer is not provided among options.

Step-by-step explanation:

To determine the magnitude and direction of the resultant displacement, let's consider each leg of the journey as a vector and then combine these vectors to find the total displacement. Initially, you walk 12.5 km south, then 8.9 km west, and finally 5 km north. To find the resultant displacement, we must subtract the northward displacement from the southward displacement to get the southward component, and take the entire westward displacement as the westward component since there is no eastward displacement to counter it.

The southward component is 12.5 km - 5 km = 7.5 km. The westward component is the full 8.9 km. We then apply the Pythagorean theorem to these perpendicular components to find the magnitude of the resultant displacement:

R = √(7.5 km)^2 + (8.9 km)^2 ≈ √(56.25 km^2 + 79.21 km^2) ≈ √(135.46 km^2) ≈ 11.6 km.

The direction is found by calculating the arctangent of the ratio between the southward and westward components:

θ = arctan(Δsouth/Δwest) = arctan(7.5/8.9) ≈ 40 degrees south of west.

Therefore, the resultant displacement is approximately 11.6 km in direction 40 degrees south of west.

Hence, the answer is not provided among options.