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?
Details and assumptions
The mass density and relative proportions of the airplane are unchanged - it's just bigger.Assume the wings work solely by the Bernoulli effect.
To find the ratio of the new velocity v1 to the original velocity v0, we need to understand the relationship between velocity and the physical characteristics of the airplane.
The Bernoulli effect states that as the speed of a fluid (such as air) increases, the pressure it exerts decreases. In the case of an airplane, this effect is utilized by the shape of the wings to generate lift.
When an airplane flies at a constant speed, the pressure difference above and below the wings allows it to stay in the air. This pressure difference is directly related to the velocity of the airplane.
Now, let's consider the situation where the airplane is scaled up by a factor of two in length, width, and height.
When the size of the airplane is scaled up, it effectively increases the surface area of the wings. This larger surface area requires more air to flow over it to generate the same amount of lift as the smaller airplane.
To maintain the same pressure difference and generate the same lift, the larger airplane needs to travel at a higher velocity.
So, in this case, if the original airplane was flying at velocity v0, the larger scaled-up airplane can only fly above a velocity v1.
To find the ratio v1/v0, we need to consider the relationship between velocity and surface area. The lift generated by the wings is directly proportional to the surface area, and inversely proportional to the velocity.
Since the size of the airplane has doubled in all dimensions, the surface area of the wings has increased by a factor of 2 * 2 = 4.
To compensate for this increased surface area, the larger airplane needs to increase its velocity by a factor of 4 to generate the same lift.
Therefore, v1/v0 = 4.
In conclusion, the new velocity v1 is four times the original velocity v0 when the size of the airplane is scaled up by a factor of two in all dimensions.