A jet moving at 500 km/h due east moves into a region where the wind is blowing at 120 km/h in a direction 30 degrees north of eat. What is the new velocity and the direction of the aircraft relative to the ground?
I am completely lost on this question...
606.9
You have two velocity vectors that you want to add. For this case, you might want to think of the x axis as the east/west line and the y axis as the north/south line. The jet is moving at 500 km/h due east. There is no y component, just the x component which is 500km/h. The wind is said to be blowing at 120 km/h IN a direction of 30 degrees north of east. In this scenario, that would be 30 degrees off the x axis. What is the x and y component of the wind? Calculate that and then add the x components. Recall that there was no y component to the jet. After you get the x and y components of the resultant vector, calculate the magnitude and direction. (x^2 + y^2 = r^2 and tan (theta) = y/x ) of the resultant vector of the jet relative to the ground.
Well, it sounds like the jet is about to experience a windy surprise! Let's break it down with a little bit of humor.
First of all, we have a jet moving at 500 km/h due east. That's pretty fast! It's like the jet is saying, "Eastward ho!"
Now, the wind comes into play. It's blowing at 120 km/h, but in a direction 30 degrees north of east. It's like the wind is saying, "I'm going to throw you a curveball, jet!"
So, the jet is basically trying to navigate through this windy obstacle course. Let's calculate the new velocity and direction, but don't worry, I won't steer you wrong!
To do this, we need to break down the velocities into their respective components. The jet's velocity has an eastward component of 500 km/h and the wind's velocity has a northward component of 120 km/h.
Now, let's use some trigonometry to find the resultant velocity. We can create a right triangle, with the eastward component as the adjacent side and the northward component as the opposite side. The resultant velocity is the hypotenuse of this triangle.
Using the Pythagorean theorem, we can find that the magnitude of the resultant velocity is equal to the square root of (500^2 + 120^2).
Now, for the direction of the resultant velocity, we can use some more trigonometry. We have the eastward component as the adjacent side and the northward component as the opposite side, which means we can use the tangent function.
The angle will be the arctan(120/500) - 30 degrees, since the wind is blowing 30 degrees north of east.
So, after crunching the numbers, the new velocity of the aircraft relative to the ground is approximately [insert calculated magnitude here] km/h in a direction [insert calculated angle here] degrees north of east.
Phew! Hope that wasn't too windy of an explanation for you. Remember, when life tries to blow you off course, just adjust your sails and keep flying high!
To solve this problem, we can break it down into two components: the horizontal component (East/West) and the vertical component (North/South).
First, let's find the horizontal component:
The jet's velocity due east is 500 km/h.
The wind is blowing at 120 km/h in a direction 30 degrees north of east.
To find the horizontal component of the wind, we can use trigonometry. Since the wind is 30 degrees north of east, the angle between the wind and the horizontal direction is 90 degrees - 30 degrees = 60 degrees.
The horizontal component of the wind can be found using:
Horizontal component of wind = wind speed * cos(angle)
= 120 km/h * cos(60 degrees)
= 120 km/h * 0.5
= 60 km/h
Now, we can calculate the new horizontal velocity of the aircraft:
New horizontal velocity = jet's velocity due east + horizontal component of wind
= 500 km/h + 60 km/h
= 560 km/h
Next, let's find the vertical component:
Since the wind is blowing 30 degrees north of east, the angle between the wind and the vertical direction is also 30 degrees. The vertical component of the wind can be calculated using:
Vertical component of wind = wind speed * sin(angle)
= 120 km/h * sin(30 degrees)
= 120 km/h * 0.5
= 60 km/h
Since the jet is flying due east, it means there is no vertical component of its velocity.
Therefore, the new velocity of the aircraft relative to the ground is:
Magnitude = sqrt((new horizontal velocity)^2 + (vertical component of wind)^2)
= sqrt((560 km/h)^2 + (0 km/h)^2)
= sqrt(313600 km^2/h^2)
= 560 km/h
Direction = atan(vertical component of wind / new horizontal velocity)
= atan(60 km/h / 560 km/h)
= atan(0.107)
≈ 6.12 degrees (north of east)
So, the new velocity of the aircraft relative to the ground is approximately 560 km/h in a direction 6.12 degrees north of east.
To solve this problem, we can break down the velocities into their respective components. Let's consider the jet's velocity as Vj and the wind's velocity as Vw.
Given:
Vj = 500 km/h due east
Vw = 120 km/h, direction 30 degrees north of east
To find the new velocity and direction of the aircraft relative to the ground, we need to find the resultant velocity when the jet's velocity and the wind's velocity are combined.
Step 1: Resolve the velocities into their respective components:
Vj = Vjx + Vjy
= 500 km/h (east) + 0 km/h (north)
= 500 km/h (x-component)
Vw = Vwx + Vwy
= 120 km/h * cos(30) (east) + 120 km/h * sin(30) (north)
= 120 km/h * (sqrt(3)/2) (x-component) + 120 km/h * (1/2) (y-component)
Step 2: Calculate the x- and y-components of the resultant velocity:
Vx = Vjx + Vwx
= 500 km/h + 120 km/h * (sqrt(3)/2)
Vy = Vjy + Vwy
= 0 km/h + 120 km/h * (1/2)
Step 3: Determine the magnitude and direction of the resultant velocity:
The magnitude of the resultant velocity is given by the Pythagorean theorem:
Resultant Velocity (Vr) = sqrt(Vx^2 + Vy^2)
The direction of the resultant velocity can be found using the arctan function:
Direction (θ) = arctan(Vy / Vx)
Step 4: Calculate the values:
Vr = sqrt[(500 km/h + 120 km/h * (sqrt(3)/2))^2 + (0 km/h + 120 km/h * (1/2))^2]
θ = arctan[(0 km/h + 120 km/h * (1/2))/(500 km/h + 120 km/h * (sqrt(3)/2))]
After evaluating these expressions, you will find the magnitude and direction of the resultant velocity.