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...
(Please answer ASAP !)

Answer 1

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

$\frac{dm}{dt}$ = rate of mass ejection

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

now from above formula

$F_t = \rho a v(v) = \rho av^2$

Now by force balance equation

$F_t = F_s$

$\rho a v^2 = kx$

by rearranging the terms now

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