An airplane is traveling 775 km/h in a direction 42.5° west of north (Figure 3-31).
(a) Find the components of the velocity vector in the northerly and westerly directions.
northerly km/h
westerly km/h
(b) How far north and how far west has the plane traveled after 3.00 h?
north km
west km
N: 775 cos 42.5°
W: 775 sin 42.5°
now recall that distance = speed * time
To solve this problem, we need to break down the velocity vector of the airplane into its components in the northerly and westerly directions.
(a) Finding the components of the velocity vector:
Given:
Velocity of the airplane = 775 km/h
Angle west of north = 42.5°
To find the component in the northerly direction, we need to find the cosine of the angle:
cos(42.5°) = adjacent/hypotenuse
cos(42.5°) = northerly component/775 km/h
Therefore,
northerly component = 775 km/h * cos(42.5°)
To find the component in the westerly direction, we need to find the sine of the angle:
sin(42.5°) = opposite/hypotenuse
sin(42.5°) = westerly component/775 km/h
Therefore,
westerly component = 775 km/h * sin(42.5°)
Plugging in the values and calculating:
northerly component = 775 km/h * cos(42.5°) ≈ 564.5 km/h
westerly component = 775 km/h * sin(42.5°) ≈ 529.6 km/h
So, the components of the velocity vector in the northerly and westerly directions are:
northerly: 564.5 km/h
westerly: 529.6 km/h
(b) Finding the distance traveled north and west after 3.00 hours:
Given:
Time = 3.00 h
Velocity in northerly direction = 564.5 km/h
Velocity in westerly direction = 529.6 km/h
To find the distance traveled north, we can use the formula:
Distance = Velocity * Time
Therefore,
Distance north = Velocity in northerly direction * Time
Similarly, to find the distance traveled west:
Distance west = Velocity in westerly direction * Time
Plugging in the values and calculating:
Distance north = 564.5 km/h * 3.00 h ≈ 1693.5 km
Distance west = 529.6 km/h * 3.00 h ≈ 1588.8 km
So, after 3.00 hours, the plane has traveled approximately 1693.5 km north and 1588.8 km west.
(Note: The values are rounded to one decimal place.)