A light aircraft with a cruising speed of 200km/h, flying in still air, first flew east for 1h 15 min, and then north for 3h. If the pilot now flies directly back to the home airfield, on what bearing should the pilot fly, and how long
will the flight take?
Eastern distance = 250 km
northern distance = 600 km
looks like a right-angled triangle, and you want the hypotenuse
D^2 = 250^2 + 600^2
D = √...
= 650
base angle of that triangle :
tan^-1 θ = 600/250
θ = 67.4°
translate that into your "bearing"
To determine the bearing the pilot should fly and the duration of the flight, let's break down the problem into steps:
Step 1: Calculate the distance traveled in the eastward direction.
Given that the aircraft flew east for 1 hour 15 minutes (or 1.25 hours) at a cruising speed of 200 km/h, we can calculate the distance using the formula:
Distance (east) = Speed * Time
Distance (east) = 200 km/h * 1.25 h
Distance (east) = 250 km
Step 2: Calculate the distance traveled in the northward direction.
Given that the aircraft flew north for 3 hours at a cruising speed of 200 km/h, we can calculate the distance using the formula:
Distance (north) = Speed * Time
Distance (north) = 200 km/h * 3 h
Distance (north) = 600 km
Step 3: Calculate the total displacement of the flight.
To calculate the total displacement, we will use the Pythagorean theorem to find the resultant distance from the east and north displacements:
Total displacement = √((Distance (east))^2 + (Distance (north))^2)
Total displacement = √((250 km)^2 + (600 km)^2)
Total displacement = √(62500 km^2 + 360000 km^2)
Total displacement = √(422500 km^2)
Total displacement = 650 km
Step 4: Calculate the bearing of the flight.
To calculate the bearing angle, we can find the inverse tangent of the east displacement divided by the north displacement:
Bearing angle = atan(Distance (east) / Distance (north))
Bearing angle = atan(250 km / 600 km)
Bearing angle ≈ 23.83 degrees
Therefore, the pilot should fly on a bearing of approximately 23.83 degrees.
Step 5: Calculate the duration of the return flight.
Since the total displacement is 650 km, and the cruising speed of the aircraft is 200 km/h, we can calculate the duration using the formula:
Duration (return flight) = Total displacement / Speed
Duration (return flight) = 650 km / 200 km/h
Duration (return flight) ≈ 3.25 hours
Hence, the flight back to the home airfield will take approximately 3 hours and 15 minutes.
To determine the bearing and flight duration for the pilot's return to the home airfield, we can break down the problem into two steps:
Step 1: Finding the distance traveled in each direction
The aircraft first flew east for 1 hour and 15 minutes. To find the distance traveled in this direction, we need to convert the time to hours:
1 hour 15 minutes = 1 + (15/60) = 1.25 hours
The speed of the aircraft is given as 200 km/h. Therefore, the distance traveled east is:
Distance = Speed x Time
Distance = 200 km/h x 1.25 h = 250 km
Next, the aircraft flew north for 3 hours. Using the same formula, the distance traveled north is:
Distance = Speed x Time
Distance = 200 km/h x 3 h = 600 km
Step 2: Calculating the bearing and flight duration for the return to the home airfield
To find the bearing, we can use trigonometry. The bearing represents an angle measured clockwise from the north direction.
Let A be the starting point (home airfield), B be the point after flying east, and C be the point after flying north. To return directly back to the home airfield, the pilot needs to fly from point C to point A.
Using the distances traveled, the triangle ABC can be constructed. We can find the bearing by finding the angle between the north direction and the line AC.
To find the angle, we can use the inverse tangent (arctan) function:
Angle = arctan(Distance flown east / Distance flown north)
Angle = arctan(250 km / 600 km)
Using a calculator, the angle is approximately 22.6 degrees.
Therefore, the pilot should fly at a bearing of 22.6 degrees clockwise from the north direction.
To calculate the flight duration for the return journey, we need to find the distance between point C and the home airfield (point A), which can be found using the Pythagorean theorem:
Distance_C_to_A = √(Distance flown east^2 + Distance flown north^2)
Distance_C_to_A = √(250 km^2 + 600 km^2)
Distance_C_to_A = √(62,500 km^2 + 360,000 km^2)
Distance_C_to_A = √(422,500 km^2)
Distance_C_to_A ≈ 650.29 km
Finally, to calculate the flight duration, we divide the distance by the cruising speed of 200 km/h:
Flight duration = Distance_C_to_A / Speed
Flight duration = 650.29 km / 200 km/h
Flight duration ≈ 3.25 hours
Therefore, the pilot should fly at a bearing of approximately 22.6 degrees clockwise from the north direction, and the flight will take approximately 3.25 hours.