A plane flies due north (90° from east) with a velocity of 100 km/h for 3 hours. During this time, a steady wind blows southeast at 30 km/h at an angle of 315° from due east. After 3 hours, where will the plane’s position be relative to its starting point? Show your work.
East component of wind = (cos 31.5 x 30) = 25.578kph.
South component of wind = (sin 13.5 x 30) = 15.625kph.
The south component slows the plane, so subtract.
(100 - 15.625) = 84.375kph north speed. In 3 hrs, it proceeds (84.375 x 3) = 253.125km.
The east component blows the aircraft towards the east, so in 3 hrs., it will move (25.578 x 3) = 76.734km. in the east direction.
Now you have a right triangle, the hypotenuse representing the plane's path, the height vertical (north) being 253.125km., and the base (east) being 76.734km.
The length of the hypotenuse is the distance from the start, so sqrt.(253.125^2 + 76.734^2) = 254.28km.northeast from start.
The angle from the start will be tanL = (253.125/76.374), atn = 73.2 degrees, north of east
To find the plane's position after 3 hours, we can break down the motion of the plane into components. The plane travels due north with a velocity of 100 km/h for 3 hours, which gives us a displacement of 100 km/h * 3 h = 300 km north.
Now let's consider the effect of the wind on the plane's motion. The wind blows southeast at 30 km/h at an angle of 315° from due east. To determine the wind's eastward and northward components, we need to find the horizontal and vertical components of the wind velocity.
The angle of 315° is measured clockwise from due east, so to determine the wind's eastward component, we use the cosine of 315°:
eastward component = wind velocity * cos(angle)
= 30 km/h * cos(315°)
= 21.21 km/h
The angle of 315° is also measured clockwise from due east, so to determine the wind's northward component, we use the sine of 315°:
northward component = wind velocity * sin(angle)
= 30 km/h * sin(315°)
= -21.21 km/h
The negative sign indicates that the wind is blowing southward. Therefore, the wind's effect is to decrease the plane's northward motion by 21.21 km/h and increase its southward motion by 21.21 km/h.
To find the plane's final position relative to its starting point, we add the displacement due to its northward motion and subtract the displacement due to the wind's southward motion:
final position = initial position + displacement north - displacement south
= 0 km + 300 km - 21.21 km/h * 3 h
= 300 km - 63.63 km
= 236.37 km north
So, after 3 hours, the plane's position will be 236.37 km north of its starting point.