An airplane is flying 200mph at 50 degrees north of east. Wind velocity is 50mph due south. What is the velocity of the airplane?
V = 200mi/h @ 50o + 50mi/h @ 270o.
X = 200*cos50 + 50*cos270 = 128.6 mi/h.
Y = 200*sin50 + 50*sin270 = 103.2 mi/h.
V^2 = X^2 + Y^2 = 27,188.2
V = 165 mi/h.
To find the velocity of the airplane, we need to combine the velocity of the airplane itself with the velocity caused by the wind.
Given:
- Airplane's velocity: 200 mph with a direction of 50 degrees north of east.
- Wind velocity: 50 mph due south.
Step 1: Resolve the airplane's velocity into its eastward and northward components.
- Eastward component: 200 mph * cos(50 degrees) = approximately 128.98 mph (rounded to two decimal places).
- Northward component: 200 mph * sin(50 degrees) = approximately 152.79 mph (rounded to two decimal places).
Step 2: Add the eastward and northward components of the airplane's velocity to the wind velocity (which is acting southward).
- Eastward component: 128.98 mph.
- Northward component (opposite direction to the wind): -152.79 mph.
- Southward component (due to the wind): -50 mph.
Step 3: Add the corresponding components to get the total velocity of the airplane.
- Eastward component: 128.98 mph.
- Northward component: -152.79 mph.
- Southward component (from the wind): -50 mph.
Total velocity of the airplane:
- To find the magnitude (speed) of the total velocity, use the Pythagorean theorem:
magnitude = sqrt((eastward component)^2 + (northward component)^2).
- magnitude = sqrt((128.98 mph)^2 + (-152.79 mph)^2) = approximately 70.17 mph (rounded to two decimal places).
- To find the direction of the total velocity, use trigonometry:
direction = arctan(northward component / eastward component).
- direction = arctan(-152.79 mph / 128.98 mph) = approximately -47.3 degrees (rounded to one decimal place).
So, the velocity of the airplane is approximately 70.17 mph at an angle of -47.3 degrees (measured counterclockwise from east).
To determine the velocity of the airplane, we need to consider the effect of both the airplane's speed and direction, as well as the wind velocity. We can break down the velocities into their respective components by using trigonometry.
1. Identify and calculate the components of the airplane's velocity:
- Speed: 200 mph
- Direction: 50 degrees north of east
To find the eastward component, we can use the cosine function because the angle measured is relative to east:
Eastward component = speed * cos(angle)
Eastward component = 200 mph * cos(50 degrees)
Eastward component ≈ 200 mph * 0.6428 ≈ 128.56 mph
To find the northward component, we can use the sine function since the angle measured is relative to east:
Northward component = speed * sin(angle)
Northward component = 200 mph * sin(50 degrees)
Northward component ≈ 200 mph * 0.7660 ≈ 153.20 mph
So, the components of the airplane's velocity are approximately 128.56 mph eastward and 153.20 mph northward.
2. Determine the wind velocity:
- Speed: 50 mph
- Direction: due south
Since the wind is blowing due south, it does not have any eastward component.
3. Calculate the net velocity of the airplane:
- Eastward component: 128.56 mph
- Wind has no eastward component
- Northward component: 153.20 mph
- Wind's southward component: -50 mph (negative because it is in the opposite direction)
To get the net velocity, we need to combine the eastward and northward components:
Net velocity = √(eastward component^2 + northward component^2 + wind's southward component^2)
Net velocity = √(128.56^2 + 153.20^2 + (-50)^2)
Net velocity = √(16532.1536 + 23484.64 + 2500)
Net velocity ≈ √42416.7936 ≈ 205.93 mph (rounded to two decimal places)
Therefore, the velocity of the airplane, considering both its speed and direction along with the wind's velocity, is approximately 205.93 mph.