A plane flies 407 km east from city A to city B in 44.0 min and then 871 km south from city B to city C in 1.90 h. For the total trip, what are the (a) magnitude (in km) and (b) direction of the plane's displacement, the (c) magnitude (in km/h) and (d) direction of its average velocity, and (e) its average speed (in km/h)? Give your angles as positive or negative values of magnitude less than 180 degrees, measured from the +x direction (east)

To solve this problem, we need to break down the plane's motion into its horizontal (east-west) and vertical (north-south) components.

First, let's calculate the magnitude and direction of the plane's displacement.

(a) Magnitude of Displacement:
The magnitude of displacement is the straight-line distance between the initial and final positions. We can find it using the Pythagorean theorem.

Magnitude of Displacement = √[(eastward displacement)^2 + (southward displacement)^2]

The eastward displacement is 407 km, and the southward displacement is 871 km. Plugging these values into the formula:

Magnitude of Displacement = √[(407 km)^2 + (871 km)^2]
Magnitude of Displacement = √[(166,249 km^2 + 758,641 km^2)]
Magnitude of Displacement = √(924,890 km^2)
Magnitude of Displacement ≈ 961.7 km

So, the magnitude of the plane's displacement is approximately 961.7 km.

(b) Direction of Displacement:
The direction of displacement can be found using the tangent of the angle between the displacement vector and the positive x-axis (east).

Direction of Displacement = arctan(southward displacement / eastward displacement)

Direction of Displacement = arctan(-871 km / 407 km)
Direction of Displacement ≈ -64.4°

The negative sign indicates that the direction is measured clockwise from the positive x-axis (east).

So, the direction of the plane's displacement is approximately -64.4°.

Next, let's calculate the magnitude and direction of the plane's average velocity.

(c) Magnitude of Average Velocity:
The magnitude of average velocity is the total displacement divided by the total time taken.

Magnitude of Average Velocity = Total Displacement / Total Time Taken

The total displacement is 961.7 km (from part a) and the total time taken is 44.0 minutes + 1.90 hours.

Total Time Taken = 44.0 min + 1.90 h
Total Time Taken = 44.0/60 h + 1.90 h
Total Time Taken ≈ 1.133 h + 1.90 h
Total Time Taken ≈ 3.033 h

Magnitude of Average Velocity = 961.7 km / 3.033 h
Magnitude of Average Velocity ≈ 317.3 km/h

So, the magnitude of the plane's average velocity is approximately 317.3 km/h.

(d) Direction of Average Velocity:
The direction of average velocity is the same as the direction of displacement. Therefore, the direction of the plane's average velocity is -64.4°.

Lastly, let's calculate the average speed.

(e) Average Speed:
The average speed is the total distance traveled divided by the total time taken.

Average Speed = Total Distance Traveled / Total Time Taken

The total distance traveled is 407 km + 871 km, and the total time taken is 44.0 minutes + 1.90 hours.

Total Distance Traveled = 407 km + 871 km
Total Distance Traveled = 1278 km

Total Time Taken is already calculated as 3.033 hours (from part c).

Average Speed = 1278 km / 3.033 h
Average Speed ≈ 421.1 km/h

So, the average speed of the plane is approximately 421.1 km/h.

To summarize:
(a) Magnitude of Displacement: Approximately 961.7 km
(b) Direction of Displacement: Approximately -64.4°
(c) Magnitude of Average Velocity: Approximately 317.3 km/h
(d) Direction of Average Velocity: Approximately -64.4°
(e) Average Speed: Approximately 421.1 km/h