A liquid is poured into a vessel at rest with the hole in the wall closed. It is filled to a height H. With what horizontal acceleration should the vessel be moved so that the liquid does not come out, when the hole is opened ?

If A be the area of the hole and p, the density of the liquid, then
$$\begin{gathered} \mathrm{F}=\frac{\mathrm{\Delta P}}{\mathrm{\Delta t}}=\frac{\mathrm{\Delta }\left(\mathrm{mv}\right)}{\mathrm{\Delta t}}=\frac{\mathrm{V\Delta m}}{\mathrm{\Delta t}} \\ =\frac{\mathrm{V\Delta }\left(\mathrm{V\rho }\right)}{\mathrm{\Delta t}}=\mathrm{v\rho }\left(\frac{\mathrm{\Delta V}}{\mathrm{\Delta t}}\right) \\ = \mathrm{v\rho } [\mathrm{volurne} \mathrm{flowing}/\sec ] \\ =\mathrm{v\rho }\left[\mathrm{Av}\right]={\mathrm{v}}^2\mathrm{A\rho } \\ \mathrm{v}= \mathrm{Velocity} \mathrm{of} \mathrm{liquid} \mathrm{coming} \mathrm{out} \mathrm{ftom} \mathrm{hole}, \\ \text{ i.e., }\mathrm{V}=\sqrt{\left(2\mathrm{gh}\right)} \\ \therefore \mathrm{F}=\left(2\mathrm{gh}\right)\mathrm{A\rho } \\ \mathrm{The} \mathrm{mass} \mathrm{of} \mathrm{the} \mathrm{liquid} \mathrm{on} \mathrm{which} \mathrm{this} \mathrm{force} \mathrm{acts} \\ \mathrm{m}=\mathrm{Al\rho } \\ \text{ Now }\mathrm{a}=\frac{\mathrm{F}}{\mathrm{m}}=\frac{\left(2\mathrm{gh}\right)\mathrm{A\rho }}{\mathrm{Al\rho }}=\frac{2\mathrm{gh}}{\mathrm{l}} \end{gathered}$$