A plane is capable of flying 200 km/h. There is a wind of 90 km/h from the east. The pilot flies in a northwest direction relative to the ground for three hours from city A to city B. Find the speed of the plane relative to the ground, the heading of the plane, and the distance between city A and city B.
Cosine Law
u=sqrt(V²+v²-2•V•v•cos135°)=
=sqrt(200²+90²-2•200•90•cos135°)=
=sqrt(40000+8100 +25455.8)=271 km/h
s=u•t=271•3=813 km
Sine Law
271/sin135°=90/sinβ
sinβ=90•sin135°/271=0.235
β=13.6° (northwest, respectively west direction)
To find the speed of the plane relative to the ground, we need to consider the effect of the wind on the plane's motion.
Let's break down the velocity vectors involved:
1. Velocity of the plane relative to the air: 200 km/h in a northwest direction.
2. Velocity of the wind: 90 km/h from the east.
Now, let's find the effective velocity of the plane relative to the ground:
1. Decompose the plane's velocity relative to the air into its eastward and northward components. Since the plane is flying northwest, we can use trigonometry to determine these components. The northwest direction can be split into two equal components: north and west. Using the right triangle formed by these components, the northward velocity can be determined as follows:
northward velocity = (hypotenuse velocity) * sin(45°) = 200 km/h * sin(45°) = 141.42 km/h
Similarly, the westward velocity can be determined as:
westward velocity = (hypotenuse velocity) * cos(45°) = 200 km/h * cos(45°) = 141.42 km/h
2. Now, let's find the plane's velocity relative to the ground by considering the effect of the wind. Since the wind is blowing from the east, it will counteract the westward component of the plane's velocity. Therefore, the groundspeed can be determined by subtracting the westward velocity of the wind from the westward velocity of the plane:
groundspeed = westward velocity of the plane - westward velocity of the wind
= 141.42 km/h - 90 km/h
= 51.42 km/h
So, the speed of the plane relative to the ground is 51.42 km/h.
To find the heading of the plane, we can use the inverse tangent function to determine the angle between the plane's velocity relative to the ground (northward component) and its velocity relative to the air (northward component):
heading = arctan(northward velocity / westward velocity)
= arctan(141.42 km/h / 141.42 km/h) = arctan(1) = 45°
Therefore, the heading of the plane is 45°, which means it is flying northwest relative to the ground.
To find the distance between city A and city B, we need to multiply the groundspeed by the time traveled:
distance = groundspeed * time
= 51.42 km/h * 3 hours
= 154.26 km
So, the distance between city A and city B is 154.26 km.