On a safari, a team of naturalists sets out toward a research station located 4.0 km away in a direction 42° north of east. After traveling in a straight line for 2.3 km, they stop and discover that they have been traveling 30° north of east, because their guide misread his compass. What are the magnitude and direction (relative to due east) of the displacement vector now required to bring the team to the research station?
magnitude
To find the magnitude and direction of the displacement vector required to bring the team to the research station, we can break down the given information into components and use vector addition.
1. First, let's find the original displacement vector from the starting point to the research station. The direction is 42° north of east, which means it forms an angle of 42° with the positive x-axis. The magnitude of this displacement is 4.0 km.
2. Next, let's determine the displacement vector for the 2.3 km traveled in the wrong direction. The direction is 30° north of east, which forms an angle of 30° with the positive x-axis. The magnitude of this displacement is 2.3 km.
3. To find the new displacement vector required to reach the research station, we need to subtract the displacement that was already traveled in the wrong direction from the original displacement vector.
4. Now, let's break down the original displacement vector using trigonometry. The x-component (displacement in the east direction) can be found using the cosine function: 4.0 km * cos(42°). The y-component (displacement in the north direction) can be found using the sine function: 4.0 km * sin(42°).
5. Similarly, let's break down the displacement traveled in the wrong direction. The x-component can be found using the cosine function: 2.3 km * cos(30°). The y-component can be found using the sine function: 2.3 km * sin(30°).
6. Subtract the x- and y-components of the displacement traveled in the wrong direction from the x- and y-components of the original displacement, respectively, to get the new x-component and y-component of the required displacement.
7. Finally, use the Pythagorean theorem to find the magnitude of the new displacement vector: sqrt(new x-component^2 + new y-component^2).
8. Use trigonometry to find the direction of the new displacement vector by taking the inverse tangent of the ratio of the new y-component to the new x-component.
By following these steps, you can calculate the magnitude and direction of the displacement vector required to bring the team to the research station.
To find the magnitude of the displacement vector required to bring the team to the research station, we need to find the length of the remaining distance.
First, let's find the distance the team has traveled in the wrong direction. This can be calculated using the law of cosines:
c^2 = a^2 + b^2 - 2ab * cos(C)
where c is the distance traveled in the wrong direction, a is the initial distance traveled, b is the distance traveled in the correct direction, and C is the angle between those two distances.
a = 2.3 km
b = 4.0 km
C = 30°
c^2 = (2.3 km)^2 + (4.0 km)^2 - 2 * 2.3 km * 4.0 km * cos(30°)
c^2 = 5.29 km^2 + 16 km^2 - 18.15 km^2 * cos(30°)
c^2 = 21.29 km^2 - 18.15 km^2 * 0.866
c^2 = 21.29 km^2 - 15.7 km^2
c^2 = 5.59 km^2
c ≈ 2.365 km
The remaining distance (magnitude of the displacement vector required to reach the research station) is the difference between the initial distance and the distance traveled in the wrong direction:
remaining distance = 4.0 km - 2.365 km
remaining distance ≈ 1.635 km
Therefore, the magnitude of the displacement vector required to bring the team to the research station is approximately 1.635 km.