Final answer:
The speed of an incompressible fluid through a constriction increases by a factor equal to the square of the factor by which the diameter decreases, as dictated by the principle of continuity.
Step-by-step explanation:
The question asks what should be proven regarding the speed of an incompressible fluid through a constriction, like in a Venturi tube. The correct answer is that the speed increases by a factor equal to the square of the factor by which the diameter decreases.
According to the principle of continuity for incompressible fluids, the product of the cross-sectional area (A) and the velocity (V) at any point along a streamline is constant: Q = A1V1 = A2V2. This means when the cross-sectional area of a pipe decreases, which happens in a constriction like a Venturi tube, the velocity of the fluid must increase in order to maintain a constant flow rate.
Demonstrating this numerically, if the diameter of a tube decreases by a factor of 2, the cross-sectional area decreases by a factor of 4 (since area is proportional to the square of the diameter). Therefore, to conserve the flow rate (Q), the speed must increase by a factor of 4, i.e., the square of the factor by which the diameter decreases.