What Is the Displacement of a Couple Walking Around a 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.
(a) What is the distance they travel?

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

i know that the distance is 6.702 km, but I don't know how to find the magnitude and the direction. I know that whatever the direction is, it is north of west.

Can you help me figure out the magnitude and the direction? please!

 

[quasar987]

Hint #1: If they walks two-thirds of the way around the circular lake, they cover an angle

$$\frac{2}{3}360 \ degrees = \frac{2}{3}2\pi \ rad = \frac{4}{3}\pi \ rad$$

Hint #2: The length of a chord subtended by an angle $\theta$ is

$$2Rsin\left(\frac{\theta}{2}\right)$$

R being the radius of the circle.

 

[MathStudent]

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.
(a) What is the distance they travel?

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

i know that the distance is 6.702 km, but I don't know how to find the magnitude and the direction. I know that whatever the direction is, it is north of west.

Can you help me figure out the magnitude and the direction? please!

Set up a coordinate system, and represent the lake as a circle with it's center and the origin.
I let my +y axis be North, and my +x axis be East ( although this is arbitrary )

The couple started at the west side which corresponds to the point where the circle intersects the -x axis { (-1.6, 0) } and travels south along the circle 2/3 of the way, ( the distance you calculated is correct by the way )

Realize this distance corresponds to the arc length of the circle ( s ) from there starting point to end point.

An equation that may be useful is

$$s = r \theta$$

where s is arc length, r is the radius of the circle and theta is the angle subtended by the arc (given in radians, not degrees).

Also note that displacement is final position minus intial position and can be represented by a vector with it's initial point (the point where the couple starts walking, corresponding to point (-1.6,0) on the circle ) and terminal point at the couple's end position
hence, your job is to find the x and y coordinates of this position on the circle where the couple stopped walking.

hope this gets you started

-MS

 

[shawonna23]

For part a do I find the answer by doing this:

(2/3)*360-180=60km

For part b is the answer 0.116

 

[MathStudent]

For part a do I find the answer by doing this:

(2/3)*360-180=60km

your original answer of 6.702 km is correct

For part b is the answer 0.116

No, don't forget to change the mode of your calculator from degrees to radians!