F=ma 2024 Problem 21

Physics · mechanics

Problem

Question: A syringe is filled with water of density

\rho

and negligible viscosity. Its body is a cylinder of cross-sectional area

A_1

, which gradually tapers into a needle with cross-sectional area

A_2 \ll A_1

. The syringe is held in place and its end is slowly pushed inward by a force

F

, so that it moves with constant speed

v

. Water shoots straight out of the needle's tip. What is the approximate value of

F

? (A)

\rho {v}^2{A}_{1}

. (B)

\frac{\rho {v}^2{A}_{1}^2}{2{A}_{2}}

. (C)

\frac{\rho {v}^2{A}_{1}^2}{{A}_{2}}

. (D)

\frac{\rho {v}^2{A}_{1}^{3}}{2{A}_{2}^2}

. (E)

\frac{\rho {v}^2{A}_{1}^{3}}{{A}_{2}^2}