How Can You Determine the Third Leg of a Sailor's Voyage Using Vector Addition?

[joe215]

Homework Statement

A sailor in a small sailboat sails 2km east, then 3.4 km NE, then an additional distance in an unknown direction. Her final position is 6.68 km directly east of her starting point. Find the magnitude and direction of the third leg of her voyage.

The Attempt at a Solution

I've drawn a diagram, but I honestly don't know where to start to find the third leg of the voyage. This one is really stumping me.

Thanks!

 

[HallsofIvy]

1. Draw a line from some starting point, A, to the right (East) 2 cm (representing 2 km) long. Call the endpoint B.

2. From B, draw a second line slanting 45 degrees up (NE) 3.4 cm long (representing 3.4 km). Call the endpoint C.

3. Go back to A and draw a line to the right (same as (1)) 6.68 cm long. That line should go right over B and exend 6.68- 2.0= 4.68 cm farther. call the end point D.

4. Draw a line from C to D. That is the "third leg". You should be able to determine its angle and length from the triangel BCD.

 

[baba89]

I would suggest approaching this problem by breaking it down into smaller components and using vector addition to solve it. First, we can label the initial leg of the voyage as vector A (2km east) and the second leg as vector B (3.4km NE). We can then use trigonometry to find the magnitude and direction of vector B, which would be approximately 2.36km in distance and 35.5 degrees north of east.

Next, we can use the fact that the final position is 6.68km east of the starting point to create a triangle with sides A, B, and the unknown third leg, which we can label as vector C. Using the Pythagorean theorem, we can find the magnitude of vector C to be approximately 4.73km.

To find the direction of vector C, we can use the law of cosines to solve for the angle opposite to vector C. This would give us an angle of approximately 51.6 degrees south of east. Therefore, the magnitude and direction of the third leg of the voyage would be 4.73km and 51.6 degrees south of east, respectively.

In summary, by breaking the problem down into smaller components and using vector addition, trigonometry, and the law of cosines, we can find the magnitude and direction of the third leg of the voyage. This approach can also be applied to more complex vector problems in physics and engineering.