A small airplane has a speech of 200km/h with respect to the air. There is strong wind blowing at 77km/h at 23 north of east with respect to the Earth . A) in which direction should the plane head in order to land at an airport due north of its present location?B) what would be the plane's speech with respect to the ground ?
To determine the direction in which the plane should head in order to land at an airport due north of its present location, we need to consider the effect of both the plane's speed and the wind's speed and direction.
A) To find the direction, we can use vector addition. The plane's velocity (v_plane) is 200 km/h, and we can represent it as a vector pointing north. The wind's velocity (v_wind) is 77 km/h blowing at 23° north of east, which we can represent as a vector pointing northeast.
To find the resultant vector representing the combined velocity of the plane and the wind, we add the two vectors. The easiest way to add vectors is to break them down into their respective components.
Let's consider the x-axis as east-west and the y-axis as north-south. The plane's velocity (v_plane) has no east-west component but has a north-south component of 200 km/h. The wind's velocity (v_wind) has an east-west component of 77*cos(23°) km/h and a north-south component of 77*sin(23°) km/h.
Adding the respective components, the resultant velocity (v_resultant) is given as:
v_resultant_x = v_plane_x + v_wind_x = 0 + 77*cos(23°) km/h
v_resultant_y = v_plane_y + v_wind_y = 200 + 77*sin(23°) km/h
Now, we can find the magnitude and direction of the resultant velocity:
Magnitude of v_resultant = sqrt((v_resultant_x)^2 + (v_resultant_y)^2)
Direction (θ) of v_resultant = arctan(v_resultant_y / v_resultant_x)
By plugging in the values, we can calculate the magnitude and direction of the resultant velocity. The direction of the resultant velocity will give us the heading the plane should take to land at the airport due north of its present location.
B) To find the plane's speed with respect to the ground (v_ground), we only need to consider the magnitude of the resultant velocity obtained above.
v_ground = sqrt((v_resultant_x)^2 + (v_resultant_y)^2)
By substituting the values, we can calculate the plane's speed with respect to the ground.
Please provide the numerical values for the wind's speed and direction (if available) so we can solve the problem accurately.