A plane flies 475 km east from city A to city B in 50.0 min and then 986 km south from city B to city C in 1.00 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).
Wow, sounds like the plane was on quite the adventure! Let's break down the information to find the answers:
(a) To find the magnitude of the plane's displacement, we can use the Pythagorean theorem. The eastward distance traveled is 475 km, and the southward distance is 986 km. So, the displacement is the square root of (475^2 + 986^2) km.
(b) To find the direction of the displacement, we can use trigonometry. We'll take the arctangent of (986/475) to get the angle from the +x direction.
(c) To find the magnitude of the average velocity, we divide the total displacement by the total time taken. The total displacement is the same as in part (a), and the total time is 50 min + 60 min.
(d) To find the direction of the average velocity, we can use the same angle as in part (b) since it represents the direction of the displacement.
(e) Average speed is simply the total distance traveled divided by the total time taken.
Now, let me calculate these for you.
To find the answers to the given questions, we need to calculate the various quantities step by step:
Step 1: Calculate the displacement from City A to City B
Given: Distance = 475 km, Time = 50.0 min
To find the magnitude of displacement, we can use the formula:
Magnitude of Displacement = Distance
Magnitude of Displacement = 475 km (east)
Step 2: Calculate the displacement from City B to City C
Given: Distance = 986 km, Time = 1.00 h
To find the magnitude of displacement, we can use the formula:
Magnitude of Displacement = Distance
Magnitude of Displacement = 986 km (south)
Step 3: Calculate the total magnitude and direction of the plane's displacement
To find the total magnitude of displacement, we can use the Pythagorean theorem:
Total magnitude of displacement = sqrt((Magnitude of Displacement1)^2 + (Magnitude of Displacement2)^2)
Total magnitude of displacement = sqrt((475 km)^2 + (986 km)^2) ≈ 1091 km
To find the direction of displacement, we can use trigonometry:
Direction = arctan(Magnitude of Displacement2/Magnitude of Displacement1)
Direction = arctan(986 km/475 km) ≈ 63.3° south of east
Step 4: Calculate the average velocity
Average velocity is the displacement divided by the time taken for the entire trip.
Given: Time = 50.0 min + 1.00 h = 1.83 h
To find the magnitude of average velocity, we can use the formula:
Magnitude of Average Velocity = Total magnitude of displacement / Time
Magnitude of Average Velocity = 1091 km / 1.83 h ≈ 596.2 km/h
To find the direction of average velocity, we can use the direction of the total displacement, which is 63.3° south of east.
Step 5: Calculate the average speed
Average speed is the total distance traveled divided by the time taken for the entire trip.
Given: Distance = 475 km + 986 km = 1461 km, Time = 1.83 h
To find the average speed, we can use the formula:
Average Speed = Total Distance / Time
Average Speed = 1461 km / 1.83 h ≈ 798.9 km/h
In conclusion, the answers to the given questions are:
(a) Magnitude of displacement = 1091 km
(b) Direction of displacement ≈ 63.3° south of east
(c) Magnitude of average velocity = 596.2 km/h
(d) Direction of average velocity ≈ 63.3° south of east
(e) Average speed = 798.9 km/h