An airplane is traveling v = 850 km/h in a direction 38.5° west of north (Fig. 3-34).
Figure 3-34.
(a) Find the components of the velocity vector in the northerly and westerly directions.
1 km/h (north)
2 km/h (west)
(b) How far north and how far west has the plane traveled after 2 h?
3 km (north)
4 km (west)
a1: 850 cos 38.5
a2: 850 sin 38.5
b3: a1 answer x 2
b4: a2 answer x 2
(a) To find the components of the velocity vector in the northerly and westerly directions, we can use trigonometry.
The velocity vector can be broken down into two components: the northerly component and the westerly component.
Using the given information, we know that the magnitude of the velocity vector (v) is 850 km/h and it is traveling in a direction 38.5° west of north.
To find the northerly component, we can use the sine function:
northerly component = v * sin(angle)
northerly component = 850 km/h * sin(38.5°)
northerly component ≈ 850 km/h * 0.616
northerly component ≈ 523.6 km/h
To find the westerly component, we can use the cosine function:
westerly component = v * cos(angle)
westerly component = 850 km/h * cos(38.5°)
westerly component ≈ 850 km/h * 0.787
westerly component ≈ 669.5 km/h
Therefore, the components of the velocity vector are approximately 523.6 km/h (north) and 669.5 km/h (west).
(b) To find how far north and how far west the plane has traveled after 2 hours, we can calculate the distances using the components of the velocity vector.
The distance north is equal to the northerly component multiplied by the time:
distance north = northerly component * time
distance north = 523.6 km/h * 2 h
distance north = 1047.2 km
The distance west is equal to the westerly component multiplied by the time:
distance west = westerly component * time
distance west = 669.5 km/h * 2 h
distance west = 1339 km
Therefore, after 2 hours, the plane has traveled approximately 1047.2 km north and 1339 km west.
To find the components of the velocity vector in the northerly and westerly directions, we can use trigonometry. The velocity vector can be broken down into two components: one in the north direction and one in the west direction.
(a)
The north component of the velocity can be found using the formula:
north component = velocity * cos(angle)
Substituting the given values:
north component = 850 km/h * cos(38.5°)
Calculating this value gives us:
north component = 850 km/h * cos(38.5°) = 850 km/h * 0.79 = 671.5 km/h (approximately)
Therefore, the north component of the velocity is approximately 671.5 km/h.
The west component of the velocity can be found using the formula:
west component = velocity * sin(angle)
Substituting the given values:
west component = 850 km/h * sin(38.5°)
Calculating this value gives us:
west component = 850 km/h * sin(38.5°) = 850 km/h * 0.61 = 518.5 km/h (approximately)
Therefore, the west component of the velocity is approximately 518.5 km/h.
(b)
To find how far north and how far west the plane has traveled after 2 hours, we can multiply the respective components of velocity by the time.
Distance traveled north = north component * time
Distance traveled north = 671.5 km/h * 2 h = 1343 km
Therefore, the plane has traveled approximately 1343 km north after 2 hours.
Distance traveled west = west component * time
Distance traveled west = 518.5 km/h * 2 h = 1037 km
Therefore, the plane has traveled approximately 1037 km west after 2 hours.