An aeroplane flies from Achester to Beeton taking 7.7 hours however the return flight takes 11.4 hours. This difference is caused by a constant wind which blows at 22.8 m/s from Achester to Beeton. Assuming that the plane travels at a constant airspeed (i.e. velocity relative to the air) how far away is Beeton from Achester?
To determine the distance between Achester and Beeton, we can use the concept of relative velocity.
Let's assume that the speed of the airplane (velocity relative to the air) is v m/s. Now, considering the first leg of the flight from Achester to Beeton, we need to add the effect of the wind.
The wind speed is given as 22.8 m/s from Achester to Beeton. When the airplane is flying with the wind, the effective speed is the sum of the airplane's airspeed and the wind speed, which is v + 22.8 m/s.
Since the first leg of the flight takes 7.7 hours, we can use the equation:
Distance = Speed × Time.
Therefore, the distance from Achester to Beeton is given by:
Distance1 = (v + 22.8) × 7.7.
Now, considering the return flight from Beeton to Achester, the wind will be acting against the airplane's motion. In this case, the effective speed will be the airplane's airspeed minus the wind speed, which is v - 22.8 m/s.
Since the return flight takes 11.4 hours, the distance from Beeton to Achester can be calculated using:
Distance2 = (v - 22.8) × 11.4.
Given that the distance between Achester and Beeton is the same for both legs of the flight, we can equate Distance1 and Distance2. Solving this equation will help us find the value of v, which can then be used to calculate the distance.
Let's equate Distance1 and Distance2:
(v + 22.8) × 7.7 = (v - 22.8) × 11.4.
Now, we can solve this equation to find v:
7.7v + 7.7 × 22.8 = 11.4v - 11.4 × 22.8.
Using algebraic manipulation, we can simplify the equation:
7.7v + 7.7 × 22.8 = 11.4v - 11.4 × 22.8.
180.36 = 3.7v.
Dividing both sides by 3.7:
v = 180.36 / 3.7.
Solving this equation gives us:
v ≈ 48.7676.
Now that we have the value of v, we can calculate the distance using Distance1:
Distance1 = (v + 22.8) × 7.7.
Plugging in the value of v:
Distance1 ≈ (48.7676 + 22.8) × 7.7.
Calculating the value gives us:
Distance1 ≈ 525.378 km.
Therefore, Beeton is approximately 525.378 km away from Achester.