Question

Area of orifice is a. Due to ejection of water of density p at a constant speed, the equilibrium compression in spring is x. The speed of ejection of water is

• A. $\sqrt{\frac{XK}{pa}}$
• B. 2$\sqrt{\frac{XK}{pa}}$
• C. $\sqrt{\frac{XK}{2pa}}$
• D. 4$\sqrt{\frac{XK}{pa}}$

Answer 1

The correct option is A $\sqrt{\frac{XK}{pa}}$

Thrust force due to ejection of water on the cart is counter balanced by spring force on the cart

so we can write

$F_t=v\frac{dm}{dt}$

here we have

$dm/dt=$ rate of mass ejection

$\frac{dm}{dt}=\rho a\frac{dx}{dt}=\rho av$

now from above formula

$F_t=\rho av\left(v\right)=\rho a{v}^2$

Now by force balance equation

$\begin{array}{c}{F}_t=F_s\\ \rho a{v}^2=kx\end{array}$

by rearranging the terms now

$v=\sqrt{\frac{kx}{\rho a}}$

Hence,

option $\left(A\right)$ is correct answer.