a man starts from a reference point O moves 100m turns 60 degrees and again moves 100m and so on then displacement just before 4^th turn will be

To determine the displacement of the man just before his fourth turn, we can visualize his movements step by step. The scenario involves a series of movements at specific angles, which can be analyzed using basic principles of geometry and trigonometry.

Understanding the Movement

The man starts at point O and moves 100 meters in a straight line. After that, he turns 60 degrees and moves another 100 meters. This pattern continues for a total of three movements before the fourth turn. Let's break down each step to find the displacement.

Step-by-Step Breakdown

• First Move: The man moves 100 meters from point O to point A.
• Second Move: He turns 60 degrees and moves 100 meters to point B.
• Third Move: He turns another 60 degrees and moves 100 meters to point C.

Calculating Coordinates

To find the displacement, we can use a coordinate system. Let's assume point O is at the origin (0, 0).

• After the first move to point A (100m straight), the coordinates are (100, 0).
• For the second move to point B, we can calculate the coordinates using trigonometry:
  • The angle from the x-axis is 60 degrees.
  • Using cosine and sine:
    • x-coordinate: 100 * cos(60°) = 100 * 0.5 = 50
    • y-coordinate: 100 * sin(60°) = 100 * √3/2 ≈ 86.6
  • So, point B's coordinates are (100 + 50, 0 + 86.6) = (150, 86.6).
• For the third move to point C, the angle is now 120 degrees (60° + 60°):
  • x-coordinate: 100 * cos(120°) = 100 * (-0.5) = -50
  • y-coordinate: 100 * sin(120°) = 100 * (√3/2) ≈ 86.6
  • Thus, point C's coordinates are (150 - 50, 86.6 + 86.6) = (100, 173.2).

Finding the Displacement

Displacement is the straight-line distance from the starting point O to the final point C. We can calculate this using the distance formula:

Distance = √[(x2 - x1)² + (y2 - y1)²]

Substituting the coordinates:

Distance = √[(100 - 0)² + (173.2 - 0)²]

Distance = √[10000 + 29909.44] = √[39909.44] ≈ 199.77 meters.

Final Thoughts

Therefore, just before the fourth turn, the displacement of the man from the reference point O is approximately 199.77 meters. This example illustrates how systematic movement and basic trigonometric principles can help us understand displacement in a two-dimensional space.