A man moves in an open field such that after moving 10 m on a straight line he makes a sharp turn of 60 degree to his left .The total displacement after 8 such turns is equal to?

1. 12m
2. 15m
3. 17.32m
4. 14.14m

To solve the problem of the man's displacement after making eight turns, we need to visualize his movement in the open field. Each time he moves 10 meters and then turns 60 degrees to his left, we can break down his path into a series of vectors. Let's analyze this step by step.

Understanding the Movement

The man starts by moving straight for 10 meters. After this, he turns 60 degrees to his left and continues moving in that new direction. This process repeats for a total of eight turns.

Visualizing the Path

When he makes a turn of 60 degrees to the left, he effectively creates a series of vectors that can be represented in a circular pattern. Each segment of his movement can be thought of as a side of a regular hexagon because the total angle turned after six moves (360 degrees) would complete a full circle.

Calculating the Displacement

To find the total displacement after eight turns, we can use the concept of vector addition. The displacement after each complete cycle of six turns will return him close to his starting point, but with some net displacement due to the additional two turns.

• After 6 turns (360 degrees), he returns to a point that is effectively the same as the starting point, but he has moved 10 meters outward from the center of the hexagon.
• After the 7th turn (60 degrees), he moves in a new direction, and after the 8th turn, he moves again in a direction that is 60 degrees from the last.

Using Trigonometry

To find the resultant displacement after eight turns, we can use the cosine and sine functions to calculate the x and y components of his movement. Each segment contributes to the overall displacement based on the angle of movement.

After 8 turns, the effective angle from the original position is 480 degrees (8 turns × 60 degrees). Since 480 degrees is equivalent to 120 degrees (480 - 360), we can calculate the resultant displacement using the formula:

Resultant Displacement = Total Distance × cos(θ) for x-component

Resultant Displacement = Total Distance × sin(θ) for y-component

Here, the total distance is 10 meters multiplied by the number of segments contributing to the resultant vector. The resultant displacement can be calculated as:

Resultant Displacement = √(x² + y²)

Final Calculation

After performing the calculations, we find that the total displacement after 8 turns is approximately 17.32 meters. This is derived from the vector components calculated based on the angles and distances moved.

Thus, the answer to the question is 17.32 meters.