Math Problem Statement

Find the following for path C in the figure below: (a) the distance traveled (in m) (b) the magnitude of the displacement (in m) from start to finish (c) the displacement (in m) from start to finish

Solution

Let's analyze the image and break down the problem step by step.

Path C (shown in red) involves motion on the displacement axis (x-axis). From the figure:

  • Path C starts at x=4mx = 4 \, \text{m},
  • It travels to x=10mx = 10 \, \text{m},
  • Then it turns back to x=7mx = 7 \, \text{m}.

We can now address the questions:

(a) The distance traveled (in m):

  • Distance is the total path length covered, regardless of direction.
  1. From x=4mx = 4 \, \text{m} to x=10mx = 10 \, \text{m} (forward) → 104=6m10 - 4 = 6 \, \text{m},
  2. From x=10mx = 10 \, \text{m} back to x=7mx = 7 \, \text{m} (backward) → 107=3m10 - 7 = 3 \, \text{m}.

So, total distance = 6m+3m=9m6 \, \text{m} + 3 \, \text{m} = 9 \, \text{m}.

(b) The magnitude of the displacement (in m) from start to finish:

  • Displacement is the shortest straight-line distance between the starting and ending points.
  • Path C starts at x=4mx = 4 \, \text{m} and ends at x=7mx = 7 \, \text{m}.

So, displacement = 74=3m7 - 4 = 3 \, \text{m}.

(c) The displacement (in m) from start to finish:

  • Displacement includes the direction. Since the movement is to the right (positive x-direction), the displacement is:
    • +3m+3 \, \text{m}.

Summary:

  • (a) Distance traveled = 9m9 \, \text{m},
  • (b) Magnitude of displacement = 3m3 \, \text{m},
  • (c) Displacement = +3m+3 \, \text{m}.

Would you like more details on any step? Let me know if you have any questions!

Here are 5 related questions:

  1. What is the difference between distance and displacement?
  2. How does reversing direction affect the total distance but not the displacement?
  3. Can displacement ever be larger than the total distance?
  4. How is velocity related to displacement?
  5. What is the significance of the positive and negative signs in displacement?

Tip: In physics, distance is a scalar quantity (only magnitude), while displacement is a vector (magnitude and direction).

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Math Problem Analysis

Mathematical Concepts

Displacement
Distance
Motion in One Dimension

Formulas

Distance = Total path length covered
Displacement = Final position - Initial position

Theorems

Concept of displacement as a vector and distance as a scalar

Suitable Grade Level

Grades 9-12