A pilot flies her route in two straight-line segments. The displacement vector A for the first segment has a magnitude of 246 km and a direction 30.0o north of east. The displacement vector B for the second segment has a magnitude of 178 km and a direction due west. The resultant displacement vector is R = A + B and makes an angle è with the direction due east. Using the component method, find (a) the magnitude of R and (b) the directional angle è.
(1) Resolve travel segments into north and east components
(2) Perform a vector addition of the north and east components separately.
(3) With the east and north components of the sum vector, N and E, compute the magnitude and direction
Magnitude = sqrt(N^2 + E^2)
Direction (N of E) = tan^-1 (N/E)
To find the magnitude of the resultant displacement vector R, we can use the component method. In this method, we break down the vectors A and B into their horizontal (x) and vertical (y) components.
Let's start with vector A:
- The magnitude of vector A is 246 km.
- The direction of vector A is 30.0 degrees north of east.
To find the x-component of vector A:
- We can use the cosine function, cos(30°) = adjacent/hypotenuse.
- adjacent = cos(30°) * 246 km.
To find the y-component of vector A:
- We can use the sine function, sin(30°) = opposite/hypotenuse.
- opposite = sin(30°) * 246 km.
Now let's move on to vector B:
- The magnitude of vector B is 178 km.
- The direction of vector B is due west, which is a direction of 270 degrees (since east is 90 degrees).
To find the x-component of vector B:
- The x-component of vector B in the west direction is negative.
- We can use the cosine function, cos(270°) = adjacent/hypotenuse.
- adjacent = cos(270°) * 178 km.
To find the y-component of vector B:
- The y-component of vector B is zero since it is in the west direction.
Now, let's calculate the x and y components of both vectors:
Vector A:
- x-component of A = cos(30°) * 246 km
- y-component of A = sin(30°) * 246 km
Vector B:
- x-component of B = cos(270°) * 178 km
- y-component of B = 0 km
Next, we can add the x and y components of vectors A and B separately to find the x and y components of the resultant vector R:
- x-component of R = (x-component of A) + (x-component of B)
- y-component of R = (y-component of A) + (y-component of B)
Now that we have the x and y components of the resultant vector R, we can use the Pythagorean theorem to find the magnitude of R:
Magnitude of R = sqrt((x-component of R)^2 + (y-component of R)^2)
Finally, to find the directional angle è, we can use the inverse tangent function:
Directional angle è = atan((y-component of R) / (x-component of R))
By following these steps, you can find (a) the magnitude of R, and (b) the directional angle è using the component method.