Impact Force of an Ultra-High-Speed Water Jet Cutter

Problem Statement

Consider the physical characteristics of a "water jet cutter" used as an industrial cutting machine. This device cuts steel and rock by the enormous force generated when water is ejected from a fine nozzle at ultra-high speed.

A stream of water with density $\rho$ρ is continuously ejected horizontally (in the $+x$+x direction) at a constant speed $v$v from a nozzle of cross-sectional area $S$S. The water stream strikes the flat face of a thick rigid plate fixed perpendicular to the water stream.

After hitting the plate, the water spreads in all directions along the plate surface, so that the velocity component perpendicular to the plate (the $x$x direction) completely vanishes. The water flow is steady; gravity, air resistance, and interactions between water droplets can all be neglected.

Find the total magnitude $I$I of the impulse exerted on the plate by the water stream over a time interval $T$T.

Constraints

• Density of water: $\rho = 1.0 \times 10^3 \ \text{kg/m}^3$ρ=1.0×103 kg/m3
• Cross-sectional area of nozzle: $S = 1.25 \times 10^{-6} \ \text{m}^2$S=1.25×10−6 m2
• Speed of ejected water: $v = 640 \ \text{m/s}$v=640 m/s
• Observation time: $T = 8.0 \ \text{s}$T=8.0 s

Input Format

Give the magnitude $I \ [\text{N}\cdot\text{s}]$I [N⋅s] of the impulse as a positive integer.