An airplane of a certain density and shape flies at a constant speed. To do so, it must fly with a certain velocity v0. If the size of the airplane is scaled up in length, width, and height by a factor of two, it can only fly above a new velocity v1. What is v1/v0?
Plane mass increases by a factor of 2^3 = 8.
Plane lift increases by a factor 2^2 = 4
Plane drag increases by an unknown factor between 1 and 2. Engine power will have to increase to compenstate for increased prssure drag and boundary layer friction.
The plane's velocity will have to increase by v1/v0 = sqrt2 so that the new lift (4*2 times higher than before) will match the increased weight (8x higher).
Elaboration on drwl's answer: since p∼ρv^2 via Bernoulli's equation
To find the ratio of the new velocity v1 to the original velocity v0, we can use the concept of scale. Let's assume that the density and shape of the airplane remain the same even when the size is scaled up.
When the length, width, and height of the airplane are all scaled up by a factor of two, the overall volume of the airplane increases by a factor of 2^3 = 8.
Since the airplane is flying at a constant speed, we can assume that the amount of air passing over it per unit time remains the same. This means that the volume of air passing over the airplane per unit time should also remain the same.
Since the volume of the scaled-up airplane is now 8 times larger, the amount of air passing over it per unit time should also increase by a factor of 8 to maintain the same flow rate.
Given that the velocity of the air passing over the airplane is v0, the new velocity v1 can be calculated as:
v1 = v0/8
Therefore, the ratio of v1 to v0 is:
v1/v0 = (v0/8)/v0 = 1/8
So, v1/v0 is equal to 1/8 or 0.125.