On a safari, a team of naturalists sets out toward a research station located 6.3 km away in a direction 42° north of east. After traveling in a straight line for 2.2 km, they stop and discover that they have been traveling 23° 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?
find magnitude and direction. show all steps!!!
To find the magnitude and direction of the displacement vector now required to bring the team to the research station, we can break it down into two components: the displacement they have already traveled (2.2 km, 23° north of east) and the remaining displacement to the research station.
Step 1: Determine the x and y components of the first displacement
The first displacement of 2.2 km, 23° north of east can be divided into its x and y components using trigonometry. Since it is north of east, the x-component will be positive and the y-component will also be positive because we are measuring angles from the positive x-axis counterclockwise.
x-component = displacement * cos(angle)
x-component = 2.2 km * cos(23°)
x-component ≈ 2.2 km * 0.921 ≈ 2.0262 km
y-component = displacement * sin(angle)
y-component = 2.2 km * sin(23°)
y-component ≈ 2.2 km * 0.389 ≈ 0.8558 km
So, the x-component of the first displacement is approximately 2.0262 km and the y-component is approximately 0.8558 km.
Step 2: Determine the remaining displacement to the research station
Since the guide misread the compass, we need to find the correct direction from the current location to the research station. The misreading was 23° north of east, so the correct direction is 23° south of east.
We can use trigonometry to find the x and y components of this remaining displacement. Since it is south of east, the x-component will be positive and the y-component will be negative.
Since the total displacement to the research station is 6.3 km, we can find the remaining displacement by subtracting the already traveled displacement (2.2 km) from the total displacement.
remaining displacement = total displacement - already traveled displacement
remaining displacement = 6.3 km - 2.2 km = 4.1 km
x-component of the remaining displacement = remaining displacement * cos(angle)
x-component of the remaining displacement = 4.1 km * cos(23°)
x-component of the remaining displacement ≈ 4.1 km * 0.921 ≈ 3.7752 km
y-component of the remaining displacement = remaining displacement * sin(angle)
y-component of the remaining displacement = 4.1 km * sin(23°)
y-component of the remaining displacement ≈ 4.1 km * 0.389 ≈ 1.5789 km
So, the x-component of the remaining displacement is approximately 3.7752 km and the y-component is approximately -1.5789 km.
Step 3: Find the total displacement vector
To find the magnitude and direction of the total displacement vector, we need to add the x-components and y-components of the already traveled displacement and the remaining displacement.
x-component of the total displacement = x-component of the first displacement + x-component of the remaining displacement
x-component of the total displacement = 2.0262 km + 3.7752 km
x-component of the total displacement ≈ 5.8014 km
y-component of the total displacement = y-component of the first displacement + y-component of the remaining displacement
y-component of the total displacement = 0.8558 km + (-1.5789 km)
y-component of the total displacement ≈ -0.7231 km
Step 4: Find the magnitude and direction of the total displacement vector
The magnitude of the total displacement vector can be determined using the Pythagorean theorem:
magnitude of total displacement = √(x-component^2 + y-component^2)
magnitude of total displacement = √(5.8014 km^2 + (-0.7231 km)^2)
magnitude of total displacement ≈ √(33.6540 km^2 + 0.5245 km^2)
magnitude of total displacement ≈ √34.1785 km^2
magnitude of total displacement ≈ 5.849 km
The direction of the total displacement vector relative to due east can be found using trigonometry:
direction = tan^(-1)(y-component / x-component)
direction = tan^(-1)(-0.7231 km / 5.8014 km)
direction ≈ tan^(-1)(-0.1247)
direction ≈ -7.125° (rounded to one decimal place)
Therefore, the magnitude of the displacement vector necessary to bring the team to the research station is approximately 5.849 km, and the direction (relative to due east) is about -7.1°.
The displacement vector required to reach the station is the original vector to the station, minus the vector of their actual initial travel.
This is a vector subtraction ptoblem.
You will have to do the steps yourself. We are not here to do your homework for you.