What is the Force of a Water Jet After Hitting a Block?

[vin300]

This is a simple question that I should've known, but I'll ask. Fluid jet with density $ flows with velocity V1 before it strikes a block, then goes in two perpendicular directions, like left, right with the same velocity V2 in opposite directions. Find the force on the block. My try: The decrease in kinetic energy density is converted into excess pressure energy, so it should be, but I'm a silly unsure person. Initial energy/vol = $V1*V1/2, Final energy= twice $ V2*V2/2 since there are two final flows. So the difference multiplied area of impact, force. $(V*V1/2 - V2*V2)*area = force. But the correct answer, they say, is $(V1*V1 -V2*V2)* area.

 

[stockzahn]

The jet transports the mass flow m' with the velocity v₁. Defining the direction of the jet in the first place in x-direction, after hitting the block, all the water flows in y-direction (perpendicular, so no component in x-direction). The force (in x-direction) is the difference of the mass flow's velocity in x-direction:

F_x = m' ⋅ (v_x2 - v_x1)

m' = ρ ⋅ v_x1 ⋅ A → F_x = ρ ⋅ A ⋅ (v_x2² - v_x1²)

v_x2² = 0 → F_x = ρ ⋅ A ⋅ (0 - v_x1²) = - ρ ⋅ A ⋅ v_x1²

That's the force on the water jet, the same force on the block, but in opposite direction: F_Bl = ρ ⋅ A ⋅ v_x1²