Final answer:
The new speed of the aircraft relative to the ground is 306.42 mi/h and the new direction is 9.46° north of east.
Step-by-step explanation:
To find the new speed and direction of the aircraft relative to the ground, we need to consider the velocities of both the plane and the wind. The plane's velocity relative to the ground is the sum of its initial velocity and the wind velocity. The wind velocity can be broken down into its north and east components using trigonometry. By adding the north components and east components separately, we can find the new speed and direction of the aircraft relative to the ground.
The north component of the wind velocity is 100 mi/h * sin(30°) = 50 mi/h, and the east component is 100 mi/h * cos(30°) = 86.6 mi/h. The east velocity of the plane remains unchanged at 300 mi/h. To find the new speed, we can use the Pythagorean theorem: speed = sqrt((east velocity)^2 + (north velocity)^2). Substituting the values, we get the new speed as sqrt((300 mi/h)^2 + (50 mi/h)^2) = 306.42 mi/h.
The direction of the new velocity can be found using the tangent function: direction = arctan(north velocity / east velocity). Substituting the values, we get the direction as arctan(50 mi/h / 300 mi/h) = 9.46° north of east. Therefore, the new speed of the aircraft relative to the ground is 306.42 mi/h and the new direction is 9.46° north of east.