Calculation of force exerted by a water jet on a moving plate

Given parameters:
- Area of water jet, A = 10 cm2 = 0.001 m2
- Velocity of water jet, Vj = 5 m/s
- Velocity of the plate, Vp = 2 m/s

1. Calculation of momentum flux of water jet
- Momentum flux is the product of density, velocity and area of the fluid
- Density of water, ρ = 1000 kg/m3
- Momentum flux of water jet, J = ρ * Vj * A
= 1000 * 5 * 0.001
= 5 Ns/m2

2. Calculation of change in momentum of water jet after striking the plate
- Change in momentum is the product of mass flow rate and velocity difference
- Mass flow rate, m = ρ * A * Vj
= 1000 * 0.001 * 5
= 5 kg/s
- Velocity difference, ΔV = Vj - Vp
= 5 - 2
= 3 m/s
- Change in momentum, ΔP = m * ΔV
= 5 * 3
= 15 Ns

3. Calculation of force exerted by water jet on the plate
- Force is the rate of change of momentum
- Force, F = ΔP / t
= (ΔP / Δt) * (Δt / t)
= J * (Δt / t)
- Δt is the time taken by water jet to strike the plate and rebound back
- Let us assume that the rebound velocity is negligible, so Δt is the time taken by water jet to come to rest
- Δt = Vj / g
= 5 / 9.81
= 0.509 s
- Total time taken by water jet to strike and rebound is twice of Δt
- t = 2 * Δt
= 1.018 s
- Force, F = 5 * (0.509 / 1.018)
= 2.5 N

4. Calculation of actual force exerted by water jet on the plate
- The above calculation gives the force exerted by water jet on a stationary plate
- In the given problem, the plate is moving towards the nozzle with a velocity of 2 m/s
- Due to this, the actual force exerted on the plate will be higher than the calculated force
- Let us assume that the water jet is perpendicular to the plate, so the angle between the plate and the jet is 90 degrees
- Actual force, Factual = F * cosθ
= 2.5 * cos90
= 0 N (when plate is stationary)
= 2.5 * cosθ (when plate is moving)
- From the given problem, the plate is moving with a velocity of 2 m/s towards the nozzle
- Let us assume that the plate is made up of a material having a coefficient of restitution of 0.8 (typical value for metals)
- The water jet will strike the plate and rebound back with a velocity of 0.8 * 5 = 4 m/s (due to conservation of momentum)