A pilot flies her route in two straight-line segments. The displacement vector A for the first segment has a magnitude of 242 km and a direction 30.0o north of east. The displacement vector for the second segment has a magnitude of 177 km and a direction due west. The resultant displacement vector is R = A + B and makes an angle θ with the direction due east. Using the component method, find (a) the magnitude of R and (b) the directional angle θ.

Question

A pilot flies her route in two straight-line segments. The displacement vector A for the first segment has a magnitude of 242 km and a direction 30.0o north of east. The displacement vector for the second segment has a magnitude of 177 km and a direction due west. The resultant displacement vector is R = A + B and makes an angle θ with the direction due east. Using the component method, find (a) the magnitude of R and (b) the directional angle θ.

Answer 1

To find the magnitude of the resultant displacement vector R, we can use the component method.

Step 1: Break down the displacement vectors A and B into their x-component and y-component.

For vector A:
Magnitude = 242 km
Direction = 30.0 degrees north of east

Using trigonometry, we can find the x-component and y-component of vector A:

A_x = magnitude * cos(direction)
= 242 km * cos(30.0 degrees)
≈ 209.40 km

A_y = magnitude * sin(direction)
= 242 km * sin(30.0 degrees)
≈ 121.00 km

For vector B:
Magnitude = 177 km
Direction = due west (which is 270 degrees)

Again, using trigonometry, we can find the x-component and y-component of vector B:

B_x = magnitude * cos(direction)
= 177 km * cos(270 degrees)
= 0 km

B_y = magnitude * sin(direction)
= 177 km * sin(270 degrees)
≈ -177 km

Step 2: Add the respective x-components and y-components of A and B to find the x-component and y-component of R.

R_x = A_x + B_x
= 209.40 km + 0 km
= 209.40 km

R_y = A_y + B_y
= 121.00 km + (-177 km)
= -56 km

Step 3: Use the Pythagorean theorem to find the magnitude of R.

|R| = sqrt(R_x^2 + R_y^2)
= sqrt((209.40 km)^2 + (-56 km)^2)
≈ 217.94 km

Therefore, the magnitude of R is approximately 217.94 km.

Step 4: Calculate the direction of R with respect to the direction due east.

θ = tan^(-1)(R_y / R_x)
= tan^(-1)(-56 km / 209.40 km)
≈ -15.40 degrees

Since the angle is negative, we add 180 degrees to get the angle with respect to the direction due east:

θ = -15.40 degrees + 180 degrees
≈ 164.60 degrees

Therefore, the directional angle θ of R with respect to the direction due east is approximately 164.60 degrees.