A student walks 19.4 km west and then 4.6 km south. After that, the student walks 7.1 km west. Using vector addition, the magnitude of the student's displacement is

Question

BlackTom AI · Accepted Answer

Since displacement components add, combine the westward legs into a single x-component and treat the southward leg as the y-component.

1) Relevant knowledge points and concepts
- Displacement is a vector; components along perpendicular axes add algebraically.
- The resultant magnitude of perpendicular components x and y is given by the Pythagorean theorem: |R| = sqrt(x^2 + y^2).
- Path length (total distance walked) is different from net displacement.

2) Step-by-step reasoning
- Westward motions: 19.4 km west followed by 7.1 km west add to a total westward (x) component of:
  x = 19.4 + 7.1 = 26.5 km (west).
- Southward motion: 4.6 km south is the y component (take magnitude 4.6 km).
- The net displacement vector has components (26.5 km, 4.6 km). Its magnitude is computed with the Pythagorean theorem.

3) Intermediate calculations
- Square the components:
  26.5^2 = 702.25
  4.6^2 = 21.16
- Sum the squares:
  702.25 + 21.16 = 723.41
- Take the square root:
  sqrt(723.41) ≈ 26.895... km
- Round to two decimal places: 26.90 km.

4) Why the correct answer is correct
- The two westward legs combine to a single x-component; the southward leg gives the perpendicular y-component. The magnitude of the resultant is exactly sqrt(26.5^2 + 4.6^2), which evaluates to 26.90 km when rounded to two decimal places.

5) Why the other choices are incorrect
- Option b (31.10) equals the total distance walked (19.4 + 4.6 + 7.1) — path length, not displacement.
- Options a (16.90) and c (13.13) do not match any correct combination of components or the Pythagorean result; they arise from incorrect algebraic or arithmetic combinations and so are not the net displacement.

6) Conclusion
- The net displacement magnitude is 26.90 km, which matches the provided correct answer.

A student walks 19.4 km west and then 4.6 km south. After that, the student walks 7.1 km west. Using vector addition, the magnitude of the student's displacement is数值题

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