A plane whose air speed is 200 km/h heads due North. But a 100 km/h northeast wind suddenly begins to blow. What is the resulting velocity of the plane with respect to the ground?
If by "northeast wind" you mean blowing in the NE direction, then the resultant vector
v = 200j + 50√2 i + 50√2 j
= 50√2 i + (200 + 50√2)j
|v| = 100√(5+2√2)
direction is N 14.64° E
Alter that if you meant blowing from the NE
Well, if the plane is heading due North, it's safe to say it's flying away from any trouble down South! Now, let's factor in that 100 km/h northeast wind. You know what they say, "When life gives you a northeast wind, let it push you to the northwest!" So, considering the wind is pushing the plane to the northwest, we can use some vector addition here.
By combining the airspeed of 200 km/h heading due North with the wind speed of 100 km/h pushing to the northwest, we can imagine the plane as being pulled in two directions simultaneously. That's right, it's like trying to dance the Macarena and the Riverdance at the same time!
Using a little trigonometry, we can determine that the angle between the North direction and the northeast direction is 45 degrees. So, when we calculate the resulting velocity, we get a bang-up combination of 200 km/h northward and 100 km/h to the northwest, at a 45-degree angle.
Now, I could bore you with the math, but let's just cut to the chase. By using some good ol' Pythagoras, we find that the resulting velocity of the plane with respect to the ground is approximately 244 km/h heading somewhere between North and Northwest.
So, hold on tight, because with that wind, you might end up in a place where snow cones are always in season!
To find the resulting velocity of the plane with respect to the ground, we can use vector addition.
First, we need to break down the velocities into their North and East components.
The airspeed of the plane heading due North can be represented as Vplane = 200 km/h North.
The wind velocity blowing from the northeast can be represented as Vwind = 100 km/h northeast.
To break down the wind velocity into North and East components, we can use the right triangle formed by the northeast direction.
The magnitude of the northeast wind velocity is given by the Pythagorean theorem:
Magnitude of Vwind = √(North component of Vwind)^2 + (East component of Vwind)^2
Substituting the values, we get:
100 km/h = √(North component of Vwind)^2 + (North component of Vwind)^2
100 km/h = √2(North component of Vwind)^2
Simplifying further, we get:
100 km/h = √2(North component of Vwind)
Now, to find the North component of Vwind, we can square both sides of the equation:
(100 km/h)^2 = 2(North component of Vwind)^2
10000 km^2/h^2 = 2(North component of Vwind)^2
Dividing both sides by 2, we get:
5000 km^2/h^2 = (North component of Vwind)^2
Taking the square root of both sides, we get:
√5000 km/h = North component of Vwind
Simplifying it further, we get:
70.7 km/h = North component of Vwind
So, the North component of the wind velocity is approximately 70.7 km/h.
Now, let's calculate the East component of the wind velocity. Since the wind is blowing northeast, the North and East components have the same magnitude.
So, the East component of the wind velocity is also 70.7 km/h.
Now, we can find the resulting velocity of the plane with respect to the ground by adding the vectors:
Resultant velocity = Vplane + Vwind
The North component of the resultant velocity is the sum of the North components of the plane's airspeed and the wind velocity:
North component of resultant velocity = 200 km/h + 70.7 km/h
North component of resultant velocity = 270.7 km/h North
The East component of the resultant velocity is the sum of the East components of the plane's airspeed and the wind velocity:
East component of resultant velocity = 0 km/h + 70.7 km/h
East component of resultant velocity = 70.7 km/h East
Therefore, the resulting velocity of the plane with respect to the ground is approximately 270.7 km/h North, 70.7 km/h East.
To find the resulting velocity of the plane with respect to the ground, we need to combine the plane's airspeed (velocity relative to the air) with the velocity of the wind (velocity of the air moving relative to the ground).
Let's break down the given information:
- The plane's airspeed is 200 km/h, which means it is traveling at 200 km/h relative to the air in the north direction.
- The wind is blowing at 100 km/h towards the northeast direction.
To combine these velocities, we can use vector addition.
1. Convert the wind's velocity from the northeast direction to its north and east components. Since it is blowing at a 45-degree angle, we can use basic trigonometry.
- 100 km/h * cos(45°) = 100 * 0.707 = 70.71 km/h (north component)
- 100 km/h * sin(45°) = 100 * 0.707 = 70.71 km/h (east component)
2. Add the north component of the wind velocity to the north velocity of the plane to get the resulting north velocity.
- 200 km/h (plane's airspeed) + 70.71 km/h (north component of the wind velocity) = 270.71 km/h (northward velocity)
3. Add the east component of the wind velocity to the east velocity of the plane to get the resulting east velocity.
- 0 km/h (plane's airspeed in the east direction) + 70.71 km/h (east component of the wind velocity) = 70.71 km/h (eastward velocity)
So, the resulting velocity of the plane, with respect to the ground, is approximately 270.71 km/h northward and 70.71 km/h eastward. We can represent this as a vector: (270.71 km/h, 70.71 km/h)