91Ó°ÊÓ

Displacement in physics refers to the straight-line distance and direction from an object's starting position to its ending position, regardless of the path taken. In the context of a circular motion, the couple walking three-fourths of the way around a circular lake will have a displacement that is less than the total distance they traveled around the circle. Only the final position relative to the start matters.

For this exercise, since they start at the west side of the circular lake and walk three-quarters around clockwise, their displacement is the straight line from the start to the endpoint on the circle. Think of this as slicing across the circle rather than following its edge.

• Displacement is often calculated using trigonometry or geometry, such as the Pythagorean theorem.
• It is represented by both magnitude (how far) and direction (which way).

By computing the straight-line distance between these two points, like a diagonal across a rectangle, you get the magnitude of the displacement.

The radius is a fundamental property of a circle that is crucial in calculating other measurements like circumference and area. It is defined as the distance from the center of the circle to any point along its edge. In this problem, the radius is given as 1.50 km.

Understanding the radius allows us to connect various formulae:

• The formula for the circumference, which is based on the radius, allows us to calculate part of the circular path the couple travels.
• In calculating displacement, the radius helps determine how far apart the starting and end points are when taking the direct path.

The radius is vital because it remains constant for any point around the circle, which simplifies many calculations and derives relationships in circular motion problems.

Circumference is the complete distance around the edge of a circle and is directly related to the circle's radius. For the lake that the couple walks around, knowing the circumference means they can understand the total distance if they were to walk a full loop.

The formula for calculating circumference is:

• \(C = 2\pi r\), where \(r\) is the radius of the circle.

In this case, since the radius is 1.50 km, the circumference becomes approximately 9.42 km.
Understanding circumference is essential because it helps determine the total traveled distance, which is a fraction of the circumference in this problem—exactly three-fourths in this scenario.

Direction in circular motion is crucial for determining how position changes around the circle. When the couple walks around the lake, their directional change isn't only based on clockwise or counterclockwise movement, but also on the angle relative to a reference direction.

In this exercise, they start walking due south from the western edge of the lake. Since they walk three-fourths of the circle, understanding their final position is essential for accurate displacement direction:

• The couple's displacement direction is described as relative to due east.
• Calculating the final direction involves understanding angles within a circle, such as 135 degrees, which indicates they've moved to a southeastern (or exactly southwest in terms of relative coordinate systems) position.

Direction helps in translating the positional change to a comprehensible vector, which combines length (or magnitude) with direction, like pointing due southwest.