You are a pilot on an aircraft carrier. You must fly to another aircraft carrier, now 1450 km at 45 degree of your position, moving at 56 km/hr due east. The wind is blowing from the south at 72 km/hr. Calculate the heading and airspeed needed to reach the carrier 2.5 hour after you take off.
Where will the other aircraft carrier be in 2.5hr? How far South would you drift in 2.5hr?
Who cares. This problem is impossible.
To calculate the heading and airspeed needed to reach the carrier, we need to consider the aircraft's ground speed, heading, and the effect of the wind. Let's break down the information provided:
1. Distance to the other aircraft carrier: 1450 km.
2. Relative position: The carrier is at a 45-degree angle (from your position).
3. Carrier's speed: The carrier is moving at 56 km/hr due east.
4. Wind speed: The wind is blowing from the south at 72 km/hr.
5. Time: You need to reach the carrier 2.5 hours after takeoff.
Now, let's determine the required heading and airspeed:
Step 1: Calculate the wind component
- To find the wind's effect, we need to resolve its components.
- Wind component in the east direction (crosswind): Wind speed × sin(angle between the wind direction and the desired heading)
- Wind component in the north direction (headwind or tailwind): Wind speed × cos(angle between the wind direction and the desired heading)
In this case, the wind is blowing from the south (90 degrees) and the desired heading is 45 degrees. Therefore, the crosswind component is:
Crosswind component = 72 km/hr × sin(90° - 45°) = 72 km/hr × sin(45°) = 72 km/hr × 0.7071 = 50.99 km/hr (approximately 51 km/hr)
The headwind component is:
Headwind component = 72 km/hr × cos(90° - 45°) = 72 km/hr × cos(45°) = 72 km/hr × 0.7071 = 50.99 km/hr (approximately 51 km/hr)
Step 2: Calculate the ground speed
- The ground speed is the speed of the aircraft relative to the Earth's surface.
- Ground speed (GS) = Airspeed - Wind component (headwind or tailwind)
Since we need to find the airspeed, we rearrange the formula:
Airspeed = Ground speed + Wind component (headwind or tailwind)
In this case, since the wind is blowing against the aircraft's direction (headwind), we subtract the headwind component:
Airspeed = 56 km/hr + 51 km/hr = 107 km/hr
Step 3: Calculate the heading
- The heading is the direction the aircraft must fly relative to true (not magnetic) north.
- To calculate the heading, we need to consider the relative position of the carrier and the wind.
- The desired (true) heading can be found using the law of sines. By setting up a right-angled triangle with the distance to the carrier as the hypotenuse, the adjacent side represents the distance traveled east with respect to the wind, and the opposite side is the distance traveled north with respect to the wind.
- Using the given information, we can calculate the desired heading angle using the formula:
Desired heading angle = tan^(-1)(position north with respect to wind / position east with respect to wind)
In this case, the position north with respect to wind is 0 km (since we'll be traveling east), and the position east with respect to wind is 1450 km.
Therefore, the desired heading angle would be:
Desired heading angle = tan^(-1)(0 km / 1450 km) ≈ tan^(-1)(0) ≈ 0°
Since the desired heading angle is 0°, you would need to fly due east. The airspeed required to reach the carrier 2.5 hours after takeoff would be approximately 107 km/hr.