A pilot is required to fly directly from london, united kingdom to rome, italy in 3.4 hours the displacement is 1.4 x 10 ^3km [s 43 e]. the wind wind velocity reported from the ground is 55km/h [s]. determine the required velocity of the plane relative to the air
Well, it seems like the pilot will be facing some pretty "windy" conditions (pun intended). Let's calculate the required velocity of the plane relative to the air.
To do this, we need to break down the problem into two components: the velocity of the plane with respect to the ground and the velocity of the wind.
First, let's calculate the velocity of the plane with respect to the ground. We know that the displacement is 1.4 x 10^3 km [s 43 e] and the time taken is 3.4 hours. So, the velocity of the plane with respect to the ground is:
Velocity (plane to ground) = Displacement / Time
= 1.4 x 10^3 km / 3.4 hours
≈ 411.76 km/h [s 43 e]
Next, let's determine the velocity of the wind. The wind velocity reported from the ground is 55 km/h [s]. So, the velocity of the wind is:
Velocity (wind) = 55 km/h [s]
Finally, to find the required velocity of the plane relative to the air, we subtract the velocity of the wind from the velocity of the plane with respect to the ground:
Velocity (plane to air) = Velocity (plane to ground) - Velocity (wind)
= 411.76 km/h [s 43 e] - 55 km/h [s]
≈ 356.76 km/h [s 43 e]
So, the required velocity of the plane relative to the air to complete the journey from London to Rome is approximately 356.76 km/h [s 43 e]. Good luck to the pilot in battling those gusty Italian winds!
To determine the required velocity of the plane relative to the air, we need to account for the effect of the wind.
Step 1: Convert the given wind velocity to the same units as the displacement. Since the displacement is given in kilometers and hours, we need to convert the wind velocity from 55 km/h to kilometers per hour.
55 km/h * 1 hour = 55 km/hr
Step 2: Break down the given displacement into its components. The displacement is given as 1.4 x 10^3 km [SE]. This means the plane is flying 1.4 x 10^3 km southeast from London to Rome.
Step 3: Determine the actual distance traveled by the plane relative to the ground. Since the wind is blowing from the south, it will affect the plane's velocity in the north-south direction. We can use the Pythagorean theorem to find the actual distance traveled.
Let x be the actual north-south distance traveled by the plane relative to the ground.
Let y be the actual east-west distance traveled by the plane relative to the ground.
Using the Pythagorean theorem: (x^2 + y^2) = (1.4 x 10^3)^2
Step 4: Determine the time component for the north-south and east-west directions. Since the plane is traveling southeast, we know that the time taken to travel in the east-west direction is equal to the time taken to travel in the north-south direction. We need to find the time using the equation: t = d/v, where t is the time, d is the distance, and v is the velocity.
Let t1 be the time taken to travel in the north-south direction.
Let t2 be the time taken to travel in the east-west direction.
t1 = x / (v_plane + v_wind)
t2 = y / (v_plane)
Step 5: Since the total time taken for the entire journey is given as 3.4 hours, we can use the equation: t1 + t2 = 3.4.
Substitute the time components found in Step 4 into this equation.
x / (v_plane + v_wind) + y / v_plane = 3.4
Step 6: Rearrange the equation to solve for the required velocity of the plane relative to the air, v_plane.
x / (v_plane + 55) + y / v_plane = 3.4
Step 7: Solve the equation for v_plane.
This equation is non-linear, so solving it directly isn't straightforward. However, we can use numerical methods or a graphing calculator to approximate the value of v_plane.
To determine the required velocity of the plane relative to the air, we need to account for the effect of the wind.
First, let's break down the information given in the question:
- Displacement: The displacement is given as 1.4 x 10^3 km [s 43 e]. This means that the plane needs to travel 1.4 x 10^3 km in the southeast direction (southeast is 135 degrees from the north).
- Flight time: The pilot needs to complete the journey in 3.4 hours.
- Wind velocity: The wind velocity is reported as 55 km/h in the south direction.
To find the required velocity of the plane relative to the air, we'll use vector addition. Let's consider two vectors - the velocity of the plane relative to the air (P) and the velocity of the wind (W). The resultant vector (R) will be the sum of these two vectors.
Let's assume the required velocity of the plane relative to the air is v km/h. We need to find the magnitude and direction of v.
Using vector addition, we can say:
R = P + W
Since the resultant displacement of the plane is in the southeast direction, we can break it down into its north and east components:
R = Ns + Ew
Now we need to find the magnitude and direction of the north and east components of R.
Using trigonometry, we can calculate the north and east components of the wind velocity. Since the wind velocity is in the south direction, the north component will be 0 km/h, and the east component will be -55 km/h.
So, we can write the equation as:
R = Ns + E*(-55)
Since the resultant displacement is given as 1.4 x 10^3 km [s 43 e], we can break it down into its north and east components:
R = N*(-sin(43)) + E*cos(43)
Now we can equate the north and east components of R to find the magnitude and direction of the required velocity of the plane relative to the air.
N*(-sin(43)) + E*cos(43) = P
Substituting the values of N and E, we have:
(-sin(43)) * v = P
E * cos(43) = 55
P = 1.4 x 10^3
Solving these equations will give you the required magnitude of the plane's velocity relative to the air (v).