Answer:
a = ½ ρ A/M v₁²
Explanation:
This scenario involves fluid mechanics, where a water jet moving at a constant velocity strikes a paddle and during this interaction, the water remains motionless. We will apply Bernoulli's equation, labeling the jet before the impact as index 1 and after the impact as index 2.
P₁ + ½ ρ v₁² + ρ g h₁ = P₂ + ½ ρ v₂² + ρ g h₂
In this case, the water is at rest post-impact, hence v₂ = 0, while h₁ = h₂ remains horizontal.
P₁-P₂ = ½ ρ v₁²
ΔP = ½ ρ v₁²
Now we can use the definition of pressure as a force divided by area.
F / A = ½ ρ v₁²
F = 1/2 ρ A v₁²
The density can be defined as
ρ = m / V
The volume is
V = A l
F = ½ m / l v₁²
With the force known, we can focus on the mass accretion of the plate M leading to its acceleration.
F = M a
a = F / M
a = ½ m/M 1/l v₁²
this can also be expressed in terms of the water's density
a = ½ ρ A/M v₁²
