A wedge of mass M and dimensions as shown in the figure is lying completely inside a region (I) of zero pressure. The pressure inside region II is 1 Pa. The surfaces are smooth. The wedge is imparted an initial velocity $v_0$ so that it just travels a distance b inside region II before finally coming to rest. The value of v₀ is 

Consider an arbitrary moment when the wedge has travelled a distance x
into region II. 
The area of the top surface inside the region II = ax sec $\theta$
$\text{ Force on it }=\mathrm{axsec}\mathrm{\theta \Delta P}=\mathrm{axsec}\mathrm{\theta }[\mathrm{\Delta P}=1]$
Component of the force opposite velocity = ax sec $\mathrm{\theta }\cdot \sin \mathrm{\theta }=\mathrm{axtan}\mathrm{\theta }\text{. }$
If it further moves by dx then the work done = ax tan $\theta$ dx 
$$\begin{gathered} \therefore \frac{1}{2}M{v}_0^2=\mathrm{atan}\mathrm{\theta }{\int }_0^{\mathrm{b}} \mathrm{xdx} \\ \Rightarrow {\mathrm{v}}_0=\sqrt{\frac{\mathrm{abh}}{\mathrm{M}}} \end{gathered}$$