An airplane flies due north at 220 km/h with respect to the air. There is a wind blowing at 65 km/h to the northeast with respect to the ground. What are the plane's speed and direction with respect to the ground?
Pg=Pa+Ag
= 220N+65NE=220N+65*.707N+65*.707E
= 220N+46N+46E=266N+46E
= sqrt(266^2+46^2) speed, direction = arctan46/366 E of N
To find the plane's speed and direction with respect to the ground, we'll use vector addition.
First, let's break down the velocities into their respective components.
The plane's velocity with respect to the air is due north, so its velocity in the north direction is 220 km/h, and its velocity in the east direction is 0 km/h (since it's not moving eastward).
The wind's velocity is blowing to the northeast, so its velocity in the north direction is 65 km/h * cos(45°), and its velocity in the east direction is 65 km/h * sin(45°).
Now we can add these two vector components to find the resultant velocity.
The north component (plane's velocity with respect to the air + wind's velocity) is 220 km/h + 65 km/h * cos(45°).
The east component (wind's velocity) is 65 km/h * sin(45°).
Using these components, we can find the magnitude and direction of the resultant velocity using the Pythagorean theorem and trigonometry.
The magnitude of the resultant velocity is given by:
Resultant speed = sqrt((north component)^2 + (east component)^2).
The direction of the resultant velocity is given by:
Resultant direction = arctan(east component / north component).
Plugging in the values:
Resultant speed = sqrt((220 km/h + 65 km/h * cos(45°))^2 + (65 km/h * sin(45°))^2).
Resultant direction = arctan((65 km/h * sin(45°)) / (220 km/h + 65 km/h * cos(45°))).
Evaluating these expressions will give us the speed and direction of the plane with respect to the ground.