An airplane flies due north at 240 km/h with respect to the air. There is a wind blowing at 75 km/h to the northeast with respect to the ground. What are the plane's speed and direction with respect to the ground?
__?__km/h
__?__° east of north
To determine the plane's speed and direction with respect to the ground, we can break down the given information into its components.
First, let's consider the airplane's velocity with respect to the air. It is flying due north at 240 km/h.
Now, let's consider the wind velocity. It is blowing at 75 km/h to the northeast with respect to the ground. The direction of the wind can be broken down into north and east components. Since it is blowing northeast, the wind has equal north and east components. Using these components, we can determine that the north component of the wind is 75/sqrt(2) km/h (approximately 53.03 km/h) and the east component is also 75/sqrt(2) km/h.
To find the plane's velocity with respect to the ground, we need to add the velocity of the airplane with respect to the air to the velocity of the wind.
For the north component:
Plane's velocity north = Airplane's velocity north + Wind's velocity north
Plane's velocity north = 240 km/h + 75/sqrt(2) km/h
For the east component:
Plane's velocity east = Wind's velocity east
Plane's velocity east = 75/sqrt(2) km/h
To find the magnitude and direction of the plane's velocity with respect to the ground, we can use the Pythagorean theorem and trigonometry.
Magnitude of the plane's velocity with respect to the ground:
Plane's speed with respect to the ground = sqrt((Plane's velocity north)^2 + (Plane's velocity east)^2)
Direction of the plane's velocity with respect to the ground:
Plane's direction with respect to the ground = atan(Plane's velocity east / Plane's velocity north)
Let's do the calculations:
Magnitude of the plane's velocity with respect to the ground:
Plane's speed with respect to the ground = sqrt((240 km/h + 75/sqrt(2) km/h)^2 + (75/sqrt(2) km/h)^2)
Plane's speed with respect to the ground ≈ 248.17 km/h
Direction of the plane's velocity with respect to the ground:
Plane's direction with respect to the ground = atan((75/sqrt(2) km/h) / (240 km/h + 75/sqrt(2) km/h))
Plane's direction with respect to the ground ≈ 9.47° east of north
Therefore, the plane's speed with respect to the ground is approximately 248.17 km/h, and its direction is approximately 9.47° east of north.
To find the plane's speed and direction with respect to the ground, we can use vector addition.
Let's break down the given information:
1. The plane's velocity with respect to the air is 240 km/h due north.
2. The wind's velocity with respect to the ground is 75 km/h to the northeast.
The first step is to calculate the resultant velocity of the plane with respect to the ground:
1. The plane's velocity with respect to the ground is equal to the vector sum of its velocity with respect to the air and the wind's velocity with respect to the ground.
To determine the resultant speed, we can use the Pythagorean theorem:
Resultant speed = √(plane's velocity with respect to the air)^2 + (wind's velocity with respect to the ground)^2
Resultant speed = √(240 km/h)^2 + (75 km/h)^2
Resultant speed = √(57600 km^2/h^2 + 5625 km^2/h^2)
Resultant speed = √63225 km^2/h^2
Resultant speed ≈ 251.45 km/h (rounded to two decimal places)
To find the direction with respect to the ground, we can use trigonometry:
tan(theta) = (wind's velocity with respect to the ground) / (plane's velocity with respect to the air)
theta = tan^(-1)((wind's velocity with respect to the ground) / (plane's velocity with respect to the air))
theta = tan^(-1)(75 km/h / 240 km/h)
theta ≈ 17.74° (rounded to two decimal places)
Therefore, the plane's speed with respect to the ground is approximately 251.45 km/h, and its direction is approximately 17.74° east of north.