To solve the problem of determining the ratio of the total distance traveled to the magnitude of the net displacement, we need to break down the motion of the particle step-by-step. Let's follow the steps:

1. The particle moves 8 meters east.
2. Then it moves 15 meters north.
3. Finally, it moves 8 meters west.

Step 1: Calculate the Total Distance Travelled

• Total distance is the sum of all individual segments of movement:
• 8 \, \text{m} + 15 \, \text{m} + 8 \, \text{m} = 31 \, \text{m}

Step 2: Calculate the Net Displacement

• Displacement is the shortest distance from the initial to the final position of the particle, which forms a right triangle.
• Here, the movement east and west cancels each other out. Therefore, the net displacement occurs due to the northward movement only.
• However, analyzing the net directional movement gives a right triangle with legs as:
  • Horizontal displacement (east-west axis) = 8 \, \text{m} - 8 \, \text{m} = 0 \, \text{m}
  • Vertical displacement (north-south axis) = 15 \, \text{m}
• Thus, the net movement is vertically upwards by 15 m.
• Using the Pythagorean theorem to find the displacement:
• \text{Displacement} = \sqrt{(0)^2 + (15)^2} = 15 \, \text{m}

Step 3: Calculate the Ratio

• The ratio of total distance to net displacement is:
• \frac{31}{15}

Hence, the ratio of the total distance traveled to the net displacement is 31:15.