A small plane can travel at 200 km/h in still air. If a 50.0 km/h wind is coming from the east, determine the ground velocity of the plane if the pilot keeps the plane pointed [N].
To determine the ground velocity of the plane, we need to consider the effect of the wind on the plane's forward motion.
Given:
Speed of the plane in still air = 200 km/h
Speed of the wind = 50.0 km/h (coming from the east)
Direction of plane's motion = North (opposite to the wind's direction)
First, let's break down the velocities into their North and East components:
1. Plane's velocity:
Since the plane is pointed North and there is no wind coming from the North, the North component of the plane's velocity is its overall velocity, which is 200 km/h.
2. Wind's velocity:
Since the wind is coming from the East, its East component velocity is 50.0 km/h.
Now, to determine the ground velocity, we need to combine the North and East components of the plane's velocity and the wind's velocity. Since the North component of the plane's velocity is its overall velocity, we only need to consider the East component of the wind's velocity.
Since the plane is pointed North and the wind is coming from the East, the two velocities are perpendicular to each other and form a right triangle. We can use the Pythagorean theorem to calculate the ground velocity.
Using the Pythagorean theorem:
Ground velocity = √(North velocity^2 + East velocity^2)
Substituting the given values:
Ground velocity = √(200 km/h)^2 + (50.0 km/h)^2
Ground velocity = √(40000 km^2/h^2 + 2500 km^2/h^2)
Ground velocity = √(42500 km^2/h^2)
Ground velocity ≈ 206.155 km/h
Therefore, the ground velocity of the plane, if the pilot keeps the plane pointed North, is approximately 206.155 km/h.
we want <200cosθ,200sinθ> + <-50,0> = <0,v>
200cosθ = 50
cosθ = .25
θ = 75.5°
v = 200sinθ = 200*.968 = 193.6