One way to administer an inoculation is with a \"gun\" that shoots the vaccine t

Question

One way to administer an inoculation is with a "gun" that shoots the vaccine through a narrow opening. No needle in necessary, for the vaccine emerges with sufficient speed to pass directly into the tissue beneath the skin. The speed is high, because the vaccine (? = 1170 kg/m^3) is held in a reservoir where a high pressure pushes it out. The pressure on the surface of the vaccine in one gun is 4200000 Pa above the atmospheric pressure outside the narrow opening. The dosage is small enough that the vaccine's surface in the reservoir is nearly stationary during an inoculation. The vertical height between the vaccine's surface in the reservoir and the opening can be ignored. Find the speed at which the vaccine emerges.

? m/s

Explanation / Answer

This is an application of the Bernoulli equation, which represents conservation of energy in an incompressible fluid.

I state below the Bernoulli equation below, assuming no frictional losses:
P1/rho + v1^2/2 + g*z1 = P2/rho + v2^2/2 + g*z2

Eliminate terms associated with gravitational energy:
P1/rho + v1^2/2 = P2/rho + v2^2/2

Define states 1 and 2:
state 1: "upstream" in the reservoir under the high pressure
state 2: "downstream" in the emerging jet of vaccine at atmospheric pressure

Make assumption:
the velocity in the reservoir is negligible compared to that in the emerging jet.

Remove such a term from the Bernoulli equation:
P1/rho = P2/rho + v2^2/2

Solve for v2:
v2^2/2 = (P1 - P2)/rho

v2^2 = 2*(P1 - P2)/rho

result:
v2 = sqrt(2*(P1 - P2)/rho)

Plug in data:
P1 - P2 = 3.7e6 Pa; rho:=1060 kg/m^3;

Numerical result:
v2 = 83.55 meters/second

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