How Do You Calculate Displacement Around a Circular Lake?

[shawonna23]

One afternoon, a couple walks two-thirds of the way around a circular lake, the radius of which is 1.60 km. They start at the west side of the lake and head due south to begin with.

What are the magnitude and direction (relative to due east) of the couple's displacement?

For the direction, is the answer found by using this: (2/3)(360 degrees) - 180?

How would I find the magnitude?

 

[christinono]

One afternoon, a couple walks two-thirds of the way around a circular lake, the radius of which is 1.60 km. They start at the west side of the lake and head due south to begin with.

What are the magnitude and direction (relative to due east) of the couple's displacement?

For the direction, is the answer found by using this: (2/3)(360 degrees) - 180?

How would I find the magnitude?

Well, (2/3)(360 degrees) doesn't equal 180 degrees...

Try making a diagram. You can draw a circle on x and y axis, where the center of the circle corresponds to the origin. Place a dot on the coordinate (-1.6 km, 0), which represents the couple's initial position. If they walk 2/3 of the way, they will end up at an angle of pi/3 North of East. As for the distance, you can use simple trig.

 

[wais]

Hello! It seems like you are having trouble with a homework problem. Let's break it down and try to find a solution.

First, let's draw a diagram to visualize the situation. We have a circular lake with a radius of 1.60 km. The couple starts at the west side and walks two-thirds of the way around in a counter-clockwise direction, ending up at the south side.

To find the magnitude of the couple's displacement, we can use the Pythagorean theorem. The displacement is the straight line distance from the starting point to the ending point. In this case, it forms the hypotenuse of a right triangle, with one leg being the radius of the lake (1.60 km) and the other leg being the remaining distance the couple walked (2/3 of the circumference, or 2/3 * 2πr = 4/3πr). So, the magnitude of the displacement can be calculated as:

√(1.60^2 + (4/3πr)^2) = 2.11 km

Next, let's find the direction of the couple's displacement. To do this, we need to use trigonometry. We know that the couple started at the west side and ended at the south side, which means their displacement is in the south-west direction. To find the angle, we can use the inverse tangent function:

tanθ = (4/3πr) / 1.60

θ = tan^-1(4/3πr / 1.60) = 60.86 degrees

However, since the question asks for the direction relative to due east, we need to subtract this angle from 90 degrees (due east is perpendicular to due south). So, the direction of the couple's displacement is:

90 - 60.86 = 29.14 degrees west of due east

I hope this helps you solve your homework problem. Remember to always draw a diagram and use the appropriate mathematical formulas to find the solution. Good luck!