A sailor in a small sailboat encounters shifting winds. She sails 2.00 km east, then 3.50 km southeast, and then an additional distance in an unknown direction. Her final position is 5.80 km directly east of the starting point.
a) Find the magnitude of the third leg of the journey.
b) Find the direction of the third leg of the journey. (Answer = East of North)
Anyone mind helping please.
Break the 3 vectors into components and solve for the unknown vector.
For this choose
North is positive y
East is positive x
First vector, 2.00km east represented by (2.00km)x + 0(km)y
Second vector, 3.50km SE represented by (3.50)(cos(315))(km)x + (3.50)(sin(315))(km)y
Third vector is unknown, represent it by (a)(km)x + (b)(km)y.
All three vectors added = 5.80(km)x + 0y
so:
2.00(km)x + 3.50(cos(315))(km)x + a(km)x = 5.80(km)x
Solve for a
0(km)y + 3.50(sin(315))(km)y + b(km)y = 0(km)y
Solve for b.
The resulting angle with this method will be relative to the x axis. Adjust the answer to reflect the degrees from the y axis.
why is it not 45 degress why 315
Here the vector angles are relative to the X axis. So, east would be 0 degrees, north is 90 degrees, west is 180 degree, and south is 270 degrees. So, SE is 270 + 45 = 315. -45 degrees could also be used.
45
Dame desu
129485
To solve this problem, we need to break down the sailor's journey into components and use vectors to determine the magnitude and direction of the third leg. Here's how we can approach it:
a) Find the magnitude of the third leg of the journey:
Step 1: Draw a diagram representing the journey. Place the starting point at the origin (0,0) and plot the coordinates for each leg of the journey. Since the final position is 5.80 km directly east of the starting point, plot a point at (5.80, 0).
Step 2: Determine the components of the first two legs. The first leg is 2.00 km east, so its components are (2.00, 0). The second leg is 3.50 km southeast, which can be divided into components of (3.50*cos(45°), -3.50*sin(45°)).
Step 3: Add up the components of the first two legs to find the total displacement. The total displacement in the x-direction is the sum of the x-components, which is 2.00 + 3.50*cos(45°). The total displacement in the y-direction is the sum of the y-components, which is -3.50*sin(45°).
Step 4: Subtract the total displacement in the x-direction from the x-coordinate of the final position to find the x-component of the third leg. In this case, the x-coordinate of the final position is 5.80, so the x-component of the third leg is 5.80 - (2.00 + 3.50*cos(45°)).
Step 5: Use the Pythagorean theorem to find the magnitude of the third leg. The magnitude is the square root of the sum of the squares of the x-component and y-component of the third leg.
b) Find the direction of the third leg of the journey:
Step 6: Calculate the direction of the third leg using trigonometry. Use the arctan function to find the angle between the third leg and the north direction. Determine if the angle is east or west of north by considering the sign of the y-component of the third leg.
By following these steps, you should be able to calculate the magnitude and direction of the third leg of the journey.