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75.

[[underlined]] Appendix C. [[/underlined]]

[[underlined]] Theory of Direct-Lift Impulse Meter. [[/underlined]]

The theory of the direct-lift impulse meter is as follows:

Calling I, the momentum of the gases that strike the end of the aluminum cylinder,

m [[subscript]] c [[/subscript]] = the mass of the aluminum cylinder,

V [[subscript]] c [[/subscript]] = the initial upward velocity of the cylinder,

A [[subscript]] c [[/subscript]] = the area of cross section of the cylinder,

A [[subscript]] g [[/subscript]] = the maximum area of cross section of the suspended system comprising the gun, lead weight, and holders,

and s = the displacement of the aluminum cylinder, as obtained from the trace on the smoked glass tube, 

we have, by the principle of Conservation of Linear Momentum, for the momentum per unit area produced by the gaseous rebound,
 
I/A [[subscript]] c [[/subscript]] = m [[subscript]] c [[/subscript]] V [[subscript]] c [[/subscript]] /A [[subscript]] c [[/subscript]] = m [[subscript]] c [[/subscript]] √2gs/A [[subscript]] c [[/subscript]].

Hence the momentum communicated to the suspended system by the gaseous rebound is 

m [[subscript]] c [[/subscript]] Ag √2gs/A [[subscript]] c [[/subscript]]

and calling [[underlined]] Q [[/underlined]] the ratio of the momentum given the gun by gaseous rebound to the observed momentum of the suspended system, we have Q = m [[subscript]] c [[/subscript]] Ag √2gs/m [[subscript]] o [[/subscript]] Ac v