An airplane is flying on a bearing of 80 degrees at 540mph. A wind is blowing with a bearing of 100 degrees at 55mph. Find the component form of the velocity of the airplane.
Mathematicians do not do navigation unfortunately so I suspect they mean that the wind is heading 100 degrees rather than coming from there (the way we navigators express wind vectors)
In that case add the plane velocity vector to the wind velocity vector:
80 degrees at 540mph +100 degrees at 55mph.
by the way 80 is ten north of east
and
100 is 10 south of east or 80 east of south
East component (x) = 540 cos 10 + 55 cos 10
North component (y) = 540 cos 80 - 55 cos 80
To find the component form of the velocity of the airplane, we need to break down the velocity into its horizontal and vertical components.
Let's start by drawing a diagram to visualize the situation:
```
^
Wind |
| -------->
(55 mph) |
|
------------------------->
| Plane's velocity
|
|
```
Now, let's break down the velocities into their horizontal and vertical components:
The velocity of the airplane can be broken down into:
1. The horizontal component, which moves the airplane eastward or to the right.
2. The vertical component, which moves the airplane northward or upward.
Since the airplane is flying on a bearing of 80 degrees, we can obtain the horizontal and vertical components by using trigonometric functions.
The horizontal component = The plane's velocity * Cosine of the bearing angle of the plane
The vertical component = The plane's velocity * Sine of the bearing angle of the plane
Calculating the horizontal component:
Horizontal component = 540 mph * Cos(80 degrees)
Horizontal component = 540 mph * Cos(80°)
Horizontal component ≈ 540 mph * 0.1736
Horizontal component ≈ 93.744 mph
Calculating the vertical component:
Vertical component = 540 mph * Sin(80 degrees)
Vertical component = 540 mph * Sin(80°)
Vertical component ≈ 540 mph * 0.9848
Vertical component ≈ 532.992 mph
Therefore, the component form of the velocity of the airplane is:
Horizontal component: 93.744 mph along the eastward or right direction.
Vertical component: 532.992 mph along the northward or upward direction.
So, the component form of the velocity of the airplane is (93.744 mph, 532.992 mph).
To find the component form of the velocity of the airplane, we can break it down into its horizontal and vertical components.
First, let's find the horizontal component of the velocity. We can use the concept of vectors and trigonometry.
The horizontal component of the velocity is given by:
Vx = V * cosθ
where V is the magnitude of the velocity (540 mph) and θ is the bearing (80 degrees).
Vx = 540 * cos(80)
Next, let's find the vertical component of the velocity. Again, we can use the concept of vectors and trigonometry.
The vertical component of the velocity is given by:
Vy = V * sinθ
where V is the magnitude of the velocity (540 mph) and θ is the bearing (80 degrees).
Vy = 540 * sin(80)
Now we have the horizontal and vertical components of the velocity. Let's write the component form of the velocity:
The component form of the velocity is given by:
V = Vx i + Vy j
where Vx is the horizontal component, Vy is the vertical component, and i and j are unit vectors along the x and y axes respectively.
So the component form of the velocity of the airplane is:
V = (540 * cos(80)) i + (540 * sin(80)) j
Calculating the values of Vx and Vy, we get:
Vx ≈ 120.246 mph
Vy ≈ 535.73 mph
Therefore, the component form of the velocity of the airplane is approximately:
V ≈ 120.246 i + 535.73 j