Problem $1$:

In the jet impact experiment, three different types of vanes are used: a flat vane, a conical vane with an angle $),$ and a hemispherical vane. $V_1$ is the water velocity leaving the nozzle. Assume a steady state. The water mass flow rate is measured to be $m_1^{},$ The density of water is $,$ and the cross$-$section area of the nozzle is $A_1.$

If hand R can beignored$-$

$(1)$ Assuming friction between water and the vane can be neglected, use Bernoulli's equation to obtain the expression to determine $V_2$ for each case. $(1+1+1$ points$)$

$(2)$ Apply the conservation of linear momentum to derive the expression of determining the impact force $F_r$ on each vane. points