An aircraft has a cruising speed of 100 m/s On a particular day, a wind is blowing from the west at 75.0 m/s
If the plane is pointed due north, what would be the magnitude and direction of the velocity relative to the ground?
If the pilot wishes to have a resultant direction of due north, in what direction should the plane be pointed?
X = 75 m/s.
Y = 100 m/s.
Q1. tan A = Y/X = 100/75 = 1.33333
A = 53.1o
V=Y/sin A = 100/sin53.1 = 125.0N.[53.1o]
Q2. Direction = 90o-53.1o=36.9o W of N.
To find the magnitude and direction of the velocity of the aircraft relative to the ground, we need to take into account the effect of the wind. Let's break down the problem step by step:
1. Start by drawing a diagram to visualize the situation. Draw a coordinate system with the north direction pointing up and the east direction pointing to the right. Label the wind velocity vector as -75.0 m/s (since it blows from the west) and the velocity of the plane as +100 m/s (since it is pointed due north).
2. To calculate the resultant velocity vector, we need to add the vectors representing the wind and the aircraft's velocity. We can use vector addition.
a. Add the x-components: 0 + (-75.0) = -75.0 m/s
b. Add the y-components: 100 + 0 = 100 m/s
So, the resultant velocity vector is (-75.0 m/s, 100 m/s).
3. To find the magnitude of the resultant velocity, use the Pythagorean theorem:
magnitude = sqrt((-75.0)^2 + 100^2) = sqrt(5625 + 10000) = sqrt(15625) = 125 m/s
Therefore, the magnitude of the velocity relative to the ground is 125 m/s.
4. To find the direction of the resultant velocity, use trigonometry. The direction can be calculated as the arctangent of the y-component divided by the x-component:
direction = arctan(100 / (-75.0)) = arctan(-4/3)
In degrees, this is approximately -53.1°. Since we are using a positive y-axis pointing north, we can say that the direction is 53.1° south of due west.
So, the magnitude of the velocity relative to the ground is 125 m/s, and the direction is 53.1° south of due west.
If the pilot wishes to have a resultant direction of due north, the plane should be pointed in the opposite direction of the wind. Since the wind is blowing from the west, the plane should be pointed towards the east.