To find the velocity of the aircraft with respect to the ground, we can use vector addition. We'll break down the velocity of the aircraft into its Northward and Westward components and then add the effect of the wind's velocity to determine the final velocity with respect to the ground.
Let's define the North direction as the positive y-direction and the West direction as the positive x-direction.
Given: Aircraft speed (in still air) = 150 km/hr (due North) Wind speed = 60 km/hr (from the North-West, which is at an angle of 45 degrees to both North and West)
Step 1: Resolve the aircraft's velocity into Northward and Westward components. The Northward component (V_N) of the aircraft's velocity is simply its speed in the North direction, which is 150 km/hr.
The Westward component (V_W) of the aircraft's velocity is zero because it is not moving in the West direction.
Step 2: Find the effect of the wind's velocity on the aircraft's motion. The wind is blowing from the North-West at an angle of 45 degrees to both North and West directions. We need to resolve the wind's velocity into its Northward and Westward components.
The Northward component (V_Nw) of the wind's velocity is given by: V_Nw = Wind speed * cos(45 degrees) = 60 km/hr * cos(45 degrees) = 60 km/hr * √(2)/2 ≈ 42.42 km/hr
The Westward component (V_Ww) of the wind's velocity is given by: V_Ww = Wind speed * sin(45 degrees) = 60 km/hr * sin(45 degrees) = 60 km/hr * √(2)/2 ≈ 42.42 km/hr
Step 3: Add the effects of the aircraft's and wind's velocities to get the velocity with respect to the ground.
The Northward component of the final velocity (V_N_final) is the sum of the Northward components of the aircraft's and wind's velocities: V_N_final = V_N + V_Nw = 150 km/hr + 42.42 km/hr ≈ 192.42 km/hr (Northward)
The Westward component of the final velocity (V_W_final) is the sum of the Westward components of the aircraft's and wind's velocities: V_W_final = V_W + V_Ww = 0 km/hr + 42.42 km/hr ≈ 42.42 km/hr (Westward)
Step 4: Calculate the magnitude and direction of the final velocity with respect to the ground.
The magnitude (V_final) of the final velocity with respect to the ground is given by the Pythagorean theorem: V_final = √(V_N_final^2 + V_W_final^2) = √(192.42^2 + 42.42^2) ≈ √(37006.16 + 1794.56) ≈ √38800.72 ≈ 197 km/hr
The direction of the final velocity (θ) with respect to the ground can be found using the arctan function: θ = arctan(V_W_final / V_N_final) = arctan(42.42 / 192.42) ≈ arctan(0.2204) ≈ 12.77 degrees West of North
So, the final velocity of the aircraft with respect to the ground is approximately 197 km/hr directed 12.77 degrees West of North.