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).
a. d = 407km[0o] + 871km[270o]
X = 407 + 871*cos270 = 407 km
Y = 871*sin270 = -871 km.
D = sqrt(407^2 + (-871^2)) = 961.4 km.
b. Tan A = Y/X = -871/407 = -2.14005
A = -64.95o = 64.95o S of E. = Direction.
c. V=D/T = 961.4/(1.9+0.7333)=2.63 km/h.
d. 64.95o S of E.
To find the answers to these questions, let's break down the problem into smaller parts.
First, let's find the magnitude and direction of the plane's displacement for each leg of the trip.
(a) Magnitude of Plane's Displacement:
For the first leg, the plane travels 407 km east from city A to city B. Since distance traveled is the magnitude of displacement, the magnitude of the displacement for this leg is 407 km.
For the second leg, the plane travels 871 km south from city B to city C. Again, the displacement is equal to the distance traveled, so the magnitude of the displacement for this leg is 871 km.
To find the total magnitude of displacement, we can use the Pythagorean theorem because the two legs form a right triangle.
Total Magnitude of Displacement = √((407 km)^2 + (871 km)^2)
Now, let's find the direction of the plane's displacement.
(b) Direction of Plane's Displacement:
For the first leg, the plane is flying east, which we can define as the positive x-direction. Therefore, the direction of displacement for the first leg is 0 degrees.
For the second leg, the plane is traveling south, which we can define as the negative y-direction. Therefore, the direction of displacement for the second leg is -90 degrees.
To find the total direction of the plane's displacement, we use the tangent function of the angle:
Total Direction = tan^(-1)((magnitude of the second leg)/(magnitude of the first leg))
Now, let's find the magnitude and direction of the average velocity.
(c) Magnitude of Average Velocity:
Average velocity is defined as the displacement divided by the time taken. To find the magnitude, we need to add the magnitudes of the two legs of the trip and then divide by the total time taken.
Total Magnitude of Average Velocity = ((407 km + 871 km) / (44.0 min + 1.90 h)) * (60 min/h)
(d) Direction of Average Velocity:
Average velocity includes both magnitude and direction. To find the direction, we need to use the tangent function:
Total Direction of Average Velocity = tan^(-1)((magnitude of the second leg)/(magnitude of the first leg))
Finally, let's find the average speed.
(e) Average Speed:
Average speed is defined as the total distance traveled divided by the total time taken.
Total Average Speed = ((407 km + 871 km) / (44.0 min + 1.90 h)) * (60 min/h)
By calculating these values using the given formulas, you can find the answers to all the questions.