Final answer:
The woman's displacement from town 1 to town 2 is determined by vector addition and trigonometry, considering the straight-line distance and direction of her travel. The eastward and westward components cancel out, leaving the northward component from her second trip as the total displacement.
Step-by-step explanation:
When calculating the displacement of the woman from town 1 to town 2, we must consider the straight-line distance and direction from the starting point to the end point, irrespective of the path taken. The woman drives 10 km eastward, then 12 km due 37° north of east, and finally 10 km westward.
To find her displacement, we use vector addition. The 10 km westward trip cancels out the initial 10 km eastward trip. The resultant vector therefore lies in the direction 37° north of east, which is the direction of the second leg of the trip.
Steps for Calculation:
- Represent the woman's second trip as a vector, with components 12 × cos(37°) km eastward and 12 × sin(37°) km northward.
- Calculate each component: Eastward = 12 × cos(37°) km, Northward = 12 × sin(37°) km.
- Find her resultant displacement by using the Pythagorean theorem, as there's no eastward component left from the first and last leg of the journey.
- Finally, express the displacement vector in terms of magnitude and direction.
The northward leg of her second trip is effectively the total displacement because her eastward and westward travels cancel out. Using trigonometry, we can calculate the actual displacement.