An airplane has an airspeed of 150 km/hr. It is to make a flight in a direction of 70 degrees while there is a 25 km/hr wind from 340 degrees. What will the airplane's actual heading be?
Please Explain this Problem to me!!!!
It is adding vectors
150@someangle+ 25@(340-180)=something@70
so break these up into N, E components.
Measure angles from true NORTH.
North equation
150cosTheta+25cos(340-180)=N?*cos70
Solve for N?
East equation
150sinTheta+25sin(340-180)=E?*sin70
Finally, the third equation:
N?/E?=tanTheta
Three equations, three unknowns. Lord, it looks like fun, doesn't it?
Graphically, this is a piece of cake.
Draw the wind vector to the origin. Draw the resultant line at 70 degrees.
Now, use a compass to make an arc of 150 to find where it intersects the 70 degree ray, if the compass arc originates from the head of the wind.
Method three.
You know two sides, one angle (between wind and resulatant) SAS triangle.
Law of cosines:
150^2=H^2+25^2-2H*25*cos(340-180-70)
and you solve H from that, H is the magnitude of the speed over ground.
To solve this problem, we need to consider the vector addition of the airplane's airspeed and the wind velocity.
Let's break down the problem into two parts: the airplane's airspeed and the wind vector.
1. Airplane's Airspeed:
The airplane's airspeed is given as 150 km/hr. This is the speed at which the airplane can travel relative to still air. We can represent this velocity as a vector in the direction of the airplane's heading. Since the airplane is making a flight in a direction of 70 degrees, we can represent the airplane's airspeed vector as A, with a magnitude of 150 km/hr and a direction of 70 degrees.
2. Wind Vector:
The wind direction is given as 340 degrees, and its speed is 25 km/hr. We can represent the wind velocity vector as W, with a magnitude of 25 km/hr and a direction of 340 degrees.
Now, to find the airplane's actual heading, we need to find the vector sum of the airplane's airspeed and the wind vector.
To add two vectors, we can break them down into their x and y components and then add the components separately.
Using trigonometry, we can calculate the x and y components of both vectors:
Airplane's Airspeed (A):
Ax = 150 km/hr * cos(70 degrees)
Ay = 150 km/hr * sin(70 degrees)
Wind Vector (W):
Wx = 25 km/hr * cos(340 degrees)
Wy = 25 km/hr * sin(340 degrees)
Next, we add the x and y components separately to find the resultant vector:
Resultant Vector (R):
Rx = Ax + Wx
Ry = Ay + Wy
Finally, we find the magnitude and direction of the resultant vector R.
Magnitude:
Magnitude (R) = √(Rx^2 + Ry^2)
Direction:
Direction (R) = arctan(Ry / Rx)
The magnitude of the resultant vector R represents the actual speed of the airplane relative to the ground, and the direction of the resultant vector R represents the actual heading of the airplane considering the effect of wind.
Therefore, to find the airplane's actual heading, you need to calculate the direction (R) using the arctan function.
To solve this problem, we need to understand vector addition and trigonometry. Here's how you can approach the problem:
1. Draw a diagram: Draw a diagram representing the situation. Label the airplane's airspeed vector as A and the wind's velocity vector as W.
2. Resolve the vectors: Break down the vectors A and W into their respective components along the north/south and east/west axes. The airspeed vector A can be resolved into two components: A_north/south and A_east/west. Similarly, the wind vector W can be resolved into two components: W_north/south and W_east/west.
3. Calculate the components: The north/south component of the airspeed vector A can be calculated using the formula A_north/south = A * cosθ, where θ is the angle between the direction of the airspeed vector and the north/south axis (70 degrees in this case). Similarly, the east/west component of A can be calculated using the formula A_east/west = A * sinθ. Similarly, you can calculate the components of the wind vector W using the angle between the wind and the north/south axis (340 degrees in this case).
4. Add the components: Add the corresponding components of A and W to find the resultant components R_north/south and R_east/west. The resultant vector R can be expressed as R = sqrt((R_north/south)^2 + (R_east/west)^2).
5. Calculate the direction: The actual heading of the airplane can be found using trigonometry. The angle between the resultant vector R and the north/south axis can be calculated as θ = arctan(R_east/west / R_north/south) using inverse tangent (arctan) function.
6. Convert the angle: The angle obtained in step 5 will be measured with respect to the north/south axis. To convert it into the desired format (0-360 degrees, clockwise from the north), you need to make adjustments. If the angle is negative (clockwise from the south), add 360 degrees to it. If the angle is positive (clockwise from the north), subtract it from 360 degrees.
By following these steps, you will be able to calculate the actual heading of the airplane taking into account both airspeed and wind direction.