To find the velocity of the aircraft with respect to the ground, we need to consider the vector addition of the aircraft's velocity and the wind's velocity.
Let's break down the velocities into their horizontal (east/west) and vertical (north/south) components:
Aircraft velocity:
- Magnitude: 150 km/hr
- Direction: Due North (which corresponds to a velocity vector pointing directly upwards)
- Vertical component: 150 km/hr (directly north)
- Horizontal component: 0 km/hr (no east/west component)
Wind velocity:
- Magnitude: 60 km/hr
- Direction: North-West (45 degrees west of north)
- Vertical component: 60 km/hr * sin(45) ≈ 42.43 km/hr (northward)
- Horizontal component: 60 km/hr * cos(45) ≈ 42.43 km/hr (eastward)
To find the resultant velocity (velocity with respect to the ground), we add the corresponding components together:
Vertical component: Aircraft vertical component + Wind vertical component = 150 km/hr + 42.43 km/hr = 192.43 km/hr (northward)
Horizontal component: Aircraft horizontal component + Wind horizontal component = 0 km/hr + 42.43 km/hr = 42.43 km/hr (eastward)
Using the Pythagorean theorem, we can find the magnitude of the resultant velocity:
Resultant velocity magnitude = sqrt((Vertical component)^2 + (Horizontal component)^2) = sqrt((192.43 km/hr)^2 + (42.43 km/hr)^2) ≈ 197.88 km/hr
Therefore, the magnitude of the aircraft's velocity with respect to the ground is approximately 197.88 km/hr.
To determine the direction of the resultant velocity, we can use trigonometry:
Resultant velocity direction = atan((Vertical component) / (Horizontal component)) = atan(192.43 km/hr / 42.43 km/hr) ≈ 77.73 degrees
Since the wind blows from the North-West, the resultant velocity points towards the North-East (opposite direction of the wind).
Therefore, the velocity of the aircraft with respect to the ground is approximately 197.88 km/hr, directed towards the North-East.