An airplane whose air speed is 620km/h is supposed to fly in a straight path 35 degrees north of east. A steady 95km/h wind is blowing from the north. In what direction (angle) should the plane head?
To determine the direction (angle) the airplane should head, we need to find the resultant velocity vector. This can be done by adding the velocity vectors of the airplane and the wind.
Let's analyze the velocities:
1. Airplane velocity: The airplane's airspeed is given as 620 km/h. Since it is flying 35 degrees north of east, we can break down this velocity into its eastward (x) and northward (y) components using trigonometry.
Eastward component (x): 620 km/h * cos(35°)
Northward component (y): 620 km/h * sin(35°)
2. Wind velocity: The wind is blowing at a steady speed of 95 km/h from the north. Since it is blowing directly from the north, its velocity is entirely in the northward (y) direction.
Now, we can find the resultant velocity vector by adding the x and y components:
Resultant x-component: Airplane x-component + Wind x-component = 620 km/h * cos(35°)
Resultant y-component: Airplane y-component + Wind y-component = 620 km/h * sin(35°) + 95 km/h
Next, we can find the magnitude and direction of the resultant velocity vector:
Magnitude of the resultant velocity vector:
Magnitude = sqrt((Resultant x-component)^2 + (Resultant y-component)^2)
Direction (angle) of the resultant velocity vector:
Angle = arctan(Resultant y-component / Resultant x-component)
Using the given values and calculations, we can find the direction (angle) the plane should head.
To find the direction (angle) the plane should head, we need to consider the effect of the wind on the plane's path. We can break down this problem into two components: the component in the east-west direction and the component in the north-south direction.
1. East-West Component:
- The airspeed of the plane is given as 620 km/h.
- The plane is supposed to fly 35 degrees north of east.
- We need to find the eastward component of the plane's velocity.
Since the plane is flying 35 degrees north of east, we can find the eastward component by using the cosine of the angle:
Eastward component = Airspeed * cos(angle)
Eastward component = 620 * cos(35)
2. North-South Component:
- The wind is blowing at a steady 95 km/h from the north.
- We need to find the northward component of the wind's velocity.
Since the wind is blowing from the north, the northward component is the same as the magnitude of the wind's velocity:
Northward component = 95 km/h
Now, we can find the resultant velocity of the plane by adding the eastward component of the plane's velocity to the westward component of the wind's velocity, and the northward component of the plane's velocity to the northward component of the wind's velocity.
Resultant velocity in the east-west direction = Eastward component of plane's velocity - Westward component of wind's velocity
Resultant velocity in the north-south direction = Northward component of plane's velocity + Northward component of wind's velocity
Finally, we can calculate the direction (angle) the plane should head using the tangent of the angle:
Angle = arctan(Resultant velocity in the east-west direction / Resultant velocity in the north-south direction)
By substituting the values, you can find the direction (angle) the plane should head.