An aeroplane's compass indicates that it is headed due to North, and it airspeed indicator shows that is moving through the air at 24 0 km/h. If there is a 100 km/h wind from west to east. In what direction should the pilot head to travel due north? What will be her velocity relative to the earth?
compass heading T degrees east of north (it will be negative of course)
North component = 240 cos T
East component = 240 sin T + 100
we need ZERO east
so
240 sin T = -100
sin T = -0.4166666.....
T = -24.6 degrees which is 360 - 24.6 on the compass
speed north over ground = 240 cos 24.6
To determine the direction the pilot should head to travel due north and the velocity relative to the Earth, we need to calculate the effect of the wind on the airplane's motion.
First, let's break down the components of the airplane's motion:
1. Airplane's velocity with respect to the air (Vairplane): This can be determined from the airspeed indicator, which shows the airplane is moving through the air at 240 km/h.
2. Wind velocity (Vwind): The wind is blowing from west to east at a speed of 100 km/h.
Next, let's determine the airplane's ground velocity, which is the velocity relative to the Earth's surface. This can be found by adding the airplane's velocity with respect to the air to the wind velocity:
Ground velocity (Vground) = Vairplane + Vwind
Vground = 240 km/h (due north) + 100 km/h (from west to east)
Vground = 340 km/h (direction will be somewhere between north and north-northeast)
So, the pilot needs to head in a direction slightly to the right of due north (towards the east) to counteract the effect of the wind blowing from west to east. The exact direction will depend on specific factors such as the angle of attack required to maintain the desired ground velocity.
The velocity of the airplane relative to the Earth will be 340 km/h in the direction mentioned above.
To determine the direction the pilot should head to travel due north, we need to consider the effect of the wind.
Step 1: Calculate the wind's component in the north-south direction:
Since the wind is blowing from west to east, its component in the north-south direction is 0 km/h.
Step 2: Calculate the plane's groundspeed:
The groundspeed is the combined effect of the plane's airspeed and the wind. Since the plane is moving through the air at 240 km/h and there is a 100 km/h wind from west to east, the groundspeed can be calculated by subtracting the wind's component from the airspeed:
Groundspeed = Airspeed - Wind's component
Groundspeed = 240 km/h - 0 km/h (since the wind's component in the north-south direction is 0 km/h)
Groundspeed = 240 km/h
Therefore, the plane's groundspeed is 240 km/h.
Step 3: Determine the direction the pilot should head:
To determine the direction the pilot should head, we need to find the angle between the plane's heading and the wind direction. Since the compass indicates that the plane is headed due north, we subtract the wind direction (west to east) from the plane's heading (due north):
Direction = Plane's heading - Wind direction
Direction = 0 degrees - 90 degrees
Direction = -90 degrees
Therefore, the pilot should head in a direction of -90 degrees, which means the plane should fly due east.
Step 4: Calculate the pilot's velocity relative to the earth:
The pilot's velocity relative to the earth is the vector sum of the plane's groundspeed and the wind's velocity. Since the groundspeed is 240 km/h and the wind's velocity is 100 km/h from west to east, we can calculate the pilot's velocity relative to the earth as follows:
Velocity relative to the earth = Groundspeed + Wind's velocity
Velocity relative to the earth = 240 km/h + 100 km/h (since the wind's velocity is from west to east)
Velocity relative to the earth = 340 km/h
Therefore, the pilot's velocity relative to the earth is 340 km/h.