Cleveland is 480 km east of Chicago. A plane leaves Chicago with an air speed of 800km/h in a wind which blows [S 45 degrees W]. (A) what heading will the pilot take in order to fly with a ground speed which is directly east? (B) how long will it take to get to Cleveland?
To determine the heading the pilot should take to fly with a ground speed directly east, we need to consider the effect of the wind.
A) Heading adjustment for ground speed directly east:
Since the wind is blowing from the southwest [S 45 degrees W], it means there is a crosswind component. To counteract this crosswind, the pilot needs to adjust the heading by compensating for the wind's direction and speed.
The crosswind component can be calculated using trigonometry. The wind direction is South 45 degrees West, which is equivalent to 225 degrees clockwise from North. The wind speed is not mentioned in the question.
Assuming the wind speed is "W" (you'll need to provide this value), the crosswind component can be calculated using the formula:
Crosswind = W * sin(angle between wind direction and desired heading)
In this case, the angle between the wind direction and east (assuming east is defined as 90 degrees clockwise from North) is:
Angle = 90 degrees - 225 degrees = -135 degrees
So, if you provide the wind speed (W), we can calculate the crosswind component accordingly. With that information, the pilot should take a heading that compensates for the crosswind component, allowing for a ground speed directly east.
B) Time to reach Cleveland:
We know that Cleveland is 480 km east of Chicago. To calculate the time it takes to reach Cleveland, we need to divide the distance by the ground speed. The ground speed can be calculated by subtracting the crosswind component from the airspeed of the plane. However, since the wind speed (W) is not provided, we cannot provide a specific time in this case.
Please provide the wind speed (W), so we can calculate the crosswind component and the time it will take to reach Cleveland.
To answer these questions, we need to break down the given information and use vector addition to determine the pilot's heading and the time it takes to reach Cleveland.
(A) To fly with a ground speed that is directly east, the pilot needs to compensate for the wind and fly in a direction that counteracts the wind's effect.
Step 1: Calculate the wind's effect:
- The wind is blowing at a bearing of 45 degrees west of south (S 45 degrees W).
- Since the destination (Cleveland) is east of the origin (Chicago), the wind is blowing from the east to the west.
- This means the wind's effect is the opposite: it pushes the plane from west to east.
Step 2: Calculate the required heading:
- Subtract the wind's effect from the desired direction (east).
- Since the wind pushes from the west, subtracting the westward component from the desired eastward direction will counteract the wind's effect.
- This can be represented as a vector addition problem.
Given:
- Required heading (resultant): R
- Wind's effect (wind vector): W (westward vector)
- Desired direction (eastward vector): E
We have: R - W = E
Step 3: Calculate the heading (R):
- Since we know the magnitude of the wind's effect (480 km) and the airspeed (800 km/h), we can calculate the components of W using trigonometry.
- The wind vector W has a magnitude of 480 km and is directed towards the west (opposite of east).
- By using trigonometry (Sine and Cosine), we can calculate the westward component of W (Wx) and the southward component of W (Wy).
Wx = 480 km * cos(45 degrees) ≈ 339.4 km
Wy = 480 km * sin(45 degrees) ≈ 339.4 km
- Now we have the wind vector W, and we can substitute it in the equation: R - W = E
R - (-339.4 km) = E
R + 339.4 km = E
So the pilot will need to take a heading of E - 339.4 km to counteract the wind and fly with a ground speed that is directly east.
(B) To calculate the time to reach Cleveland, we need to use the ground speed.
Given:
- Airspeed of the plane: 800 km/h
- Ground speed = Speed of the plane (airspeed) - Effect of wind
Step 1: Calculate the ground speed:
- Subtract the wind's westward component from the airspeed to get the ground speed.
- Ground speed = 800 km/h - Wx
Ground speed = 800 km/h - 339.4 km/h ≈ 460.6 km/h
Step 2: Calculate the time:
- Divide the distance between Chicago and Cleveland (480 km) by the ground speed to get the time taken to travel that distance.
Time = Distance / Ground speed
Time = 480 km / 460.6 km/h ≈ 1.042 hours (rounded to 3 decimal places)
Therefore, it will take approximately 1.042 hours (or 1 hour and 2.5 minutes) to reach Cleveland.