A plane has a speed of 250mph and flies with a bearing of N35degE. The wind is blowing from the west at 25 mph with a heading of due east. What is the actual speed and heading of the plane? Round to one decimal place.
250 @ N35E = (143.4,204.8)
25 @ E = (25,0)
add them to get
(168.4,204.8) = 265.1 @ N39.5E
To find the actual speed and heading of the plane, we need to separate the velocity contributed by the plane's forward motion from the velocity contributed by the wind.
1. Let's start with the plane's velocity components:
- The plane's velocity in the north direction: Vn = 250 mph * sin(35 degrees) = 250 mph * 0.5736 = 143.4 mph
- The plane's velocity in the east direction: Ve = 250 mph * cos(35 degrees) = 250 mph * 0.8192 = 204.8 mph
2. Now let's consider the wind's velocity components:
- The wind's velocity in the north direction: Wn = 25 mph * sin(90 degrees) = 25 mph (since the wind is blowing from the west)
- The wind's velocity in the east direction: We = 25 mph * cos(90 degrees) = 0 mph (since the wind is blowing from the west)
3. To find the actual velocity of the plane, we add the velocity components of the plane and the wind:
- The total velocity in the north direction: Vn + Wn = 143.4 mph + 25 mph = 168.4 mph
- The total velocity in the east direction: Ve + We = 204.8 mph + 0 mph = 204.8 mph
4. To find the actual speed of the plane, we use Pythagoras' theorem:
- Actual speed = sqrt((total velocity in the north direction)^2 + (total velocity in the east direction)^2)
- Actual speed = sqrt((168.4 mph)^2 + (204.8 mph)^2) = sqrt(28339.56 mph^2 + 41943.04 mph^2) = sqrt(70282.6 mph^2) = 265.1 mph (rounded to one decimal place)
5. To find the actual heading of the plane, we use the inverse tangent function:
- Actual heading = arctan((total velocity in the north direction) / (total velocity in the east direction))
- Actual heading = arctan(168.4 mph / 204.8 mph) = arctan(0.8213) = 39.6 degrees (rounded to one decimal place)
Therefore, the actual speed of the plane is approximately 265.1 mph, and the actual heading is approximately N39.6°E.
To determine the actual speed and heading of the plane, we need to break down the given information and analyze the components separately.
First, let's consider the effect of the wind on the plane's motion. Since the wind is blowing from the west at 25 mph, it will affect both the speed and heading of the plane.
To find the effect of the wind on the speed, we can use vector addition. The wind has a velocity of 25 mph to the east, and the plane's actual velocity will be the resultant of the plane's speed (250 mph) and the wind's velocity.
Using vector addition, we can form a right triangle with the plane's speed as the hypotenuse and the wind's velocity as one of the legs. The other leg, representing the plane's actual speed, can be found using the Pythagorean theorem.
Let's calculate it:
Wind's velocity: 25 mph to the east
Plane's speed: 250 mph
By applying the Pythagorean theorem, we have:
Plane's actual speed = sqrt(250^2 - 25^2)
= sqrt(62500 - 625)
= sqrt(61875)
≈ 248.7 mph (rounded to one decimal place)
Now let's consider the effect of the wind on the plane's heading. The plane's original bearing is N35degE, and the wind is blowing directly from the west. To find the actual heading, we need to subtract the wind's direction from the plane's original bearing.
Original bearing: N35degE
Wind direction: West
To find the actual heading, we subtract the wind's direction from the plane's original bearing:
Actual heading = 35degE - 90deg = -55deg (since west is 90 degrees)
So, the actual speed of the plane is approximately 248.7 mph, and the actual heading is -55 degrees (rounded to one decimal place).
Note: The negative sign on the heading indicates that the plane's heading is west of north, or it is flying in a northwest direction.
Vpw = Vp + Vw = 250[N35oE] + 25 =
250*sin35 + i250*Cos35 + 25 =
143.4 + 204.8i + 25 = 168.4 + 204.8i =
265mi/h[N50.6oE].