One afternoon, a couple walks three-fourths of the way around a circular lake, the radius of which is 2.12 km. They start at the west side of the lake and head due south to begin with. (a) What is the distance they travel? (b) What is the magnitude of the couple’s displacement? (c) What is the direction (relative to due east) of the couple’s displacement?
To solve this problem, we can break it down into steps:
Step 1: Find the circumference of the circular lake.
Since the radius is given as 2.12 km, we can use the formula for the circumference of a circle, C = 2πr, where r is the radius. Thus, the circumference is C = 2π(2.12) km = 4.24π km.
Step 2: Find the distance traveled by the couple.
If they walk three-fourths of the way around the lake, they cover 3/4 of the circumference. So the distance traveled is (3/4) × 4.24π km = 3.18π km.
To find the numerical value, we can use the approximation for π. Let's assume π ≈ 3.14.
Therefore, the distance traveled is approximately 3.18 × 3.14 km = 9.9972 km or approximately 10 km (rounded to the nearest whole number).
So the answer to part (a) is that the couple traveled approximately 10 km.
Step 3: Find the couple's displacement.
The displacement is the straight-line distance from the starting point to the ending point. In this case, they end up on the south side of the lake, which is directly opposite their starting point on the west.
The displacement is equal to the diameter of the circle, which is twice the radius. Therefore, the displacement is 2 × 2.12 km = 4.24 km.
So the answer to part (b) is that the magnitude of the couple's displacement is 4.24 km.
Step 4: Find the direction of the couple's displacement.
To determine the direction, we need to find the angle between the displacement vector and due east.
Since the couple started on the west side and ended up on the south side of the lake, they moved in a clockwise direction. Therefore, the displacement is in the fourth quadrant.
To find the angle, we can use the inverse trigonometric function arctan. We can calculate the angle between due east and the displacement vector using the formula:
θ = arctan(disp_y / disp_x)
In this case, the displacement vector has an x-component of 0 km (since it is in the y-direction only) and a y-component of -4.24 km (since it points downward). The negative sign indicates that the angle is in the fourth quadrant.
Thus, θ = arctan((-4.24) / 0) = -90 degrees.
Therefore, the direction of the couple's displacement, relative to due east, is 90 degrees counterclockwise from due east.
So the answer to part (c) is that the direction of the couple's displacement, relative to due east, is 90 degrees counterclockwise.