A plane has a speed of p km/h in still air. It makes around trip flying with and against the wind of w km/h.Show that its average speed is p^2-w^2/p km/h
time with the wind = D/(p+w) , where D is the distance traveled
time against the wind = D/(p-w)
total time = D/(p+w) + D/(p-w)
= (D(p-w) + D(p+w))/((p+w)(p-w))
= 2Dp/(p^2 - w^2)
avg speed = total distance over total time
= 2D / (2Dp/(p^2 - w^2) )
= 2D ( p^2 - w^2)/(2Dp)
= (p^2 - w^2)/p
for a 1-km round trip, the time taken is
1/(p+w) + 1/(p-w) = 2/s
2p/(p^2-w^2) = 2/s
s = (p^2-w^2)/p
To find the average speed of the plane, we need to consider the distance traveled and the time taken for both legs of the journey.
Let's assume the distance traveled in one direction is D km.
When the plane is flying with the wind, its effective speed will be increased. So, the speed of the plane relative to the ground will be (p + w) km/h.
The time taken to travel the distance D when flying with the wind can be found using the formula:
Time = Distance / Speed
So, the time taken for the journey with the wind will be:
Time1 = D / (p + w)
Similarly, when the plane is flying against the wind, its effective speed will be decreased. So, the speed of the plane relative to the ground will be (p - w) km/h.
The time taken to travel the distance D when flying against the wind can be found using the same formula:
Time = Distance / Speed
So, the time taken for the journey against the wind will be:
Time2 = D / (p - w)
Now, let's calculate the total time taken for the round trip by adding the times taken for both legs:
Total Time = Time1 + Time2
= D / (p + w) + D / (p - w)
To find the average speed, we can use the formula:
Average Speed = Total Distance / Total Time
Since the total distance traveled in the round trip is 2D, we can substitute it in the formula:
Average Speed = 2D / (D / (p + w) + D / (p - w))
= 2 / (1 / (p + w) + 1 / (p - w))
= 2 / ((p - w + p + w) / ((p + w)(p - w)))
= 2(p + w)(p - w) / 2(p^2 - w^2)
= (p^2 - w^2) / (p^2 - w^2)
= 1
Therefore, the average speed of the plane is p^2 - w^2 / p km/h.