Since the interstellar medium is highly ionised, the physical collecting funnel can be augmented by an electromagnetic field system, which will draw nuclei into the funnel from a much wider area.

Since a ramjet gathers both its fuel and reaction mass as it goes, the standard reaction mass calculations do not apply. Some other relevant equations can be derived, however.

Δ momentum =By the conservation of momentum, this is equal to the change in momentum of the ramjet:m(v-u)

where:m(v-u) =MΔV

M= the mass of the ramjet ship

ΔV= the change in velocity of the ramjet ship.

where:a=dV/dt=m(v-u) /M dt

Now, the change in kinetic energy of the interstellar medium material Δ (dt= an "infinitesimal change in time" (I am not bothering with strict formalities of calculus here).

But (P dt= Δ (m v^{2}) / 2

=m(v^{2}-u^{2}) / 2

=m(v-u) (v+u) / 2

Now consider the volume of interstellar medium swept up by the ramjet funnel. If the effective funnel (including any electromagnetic attraction fields) is circular, with a radiusP dt=a M dt V

P=a M V

πIf the density of hydrogen nuclei in the interstellar medium is ρ (in mass per unit volume units), then the mass of hydrogen nuclei swept up in timer^{2}V dt

πThis mass is available for conversion into energy, with a nuclear fusion efficiency η (η is 0.753% for hydrogen fusion), so:r^{2}Vρdt

where:E=m c^{2}

P dt= πr^{2}Vρ ηc^{2}dt

Substituting the formula for power above and rearranging:c= the speed of light.

a= πr^{2}ρ ηc^{2}/M

This means the acceleration of a ramjet is dependent only on the size of the collecting funnel, density of the interstellar medium, efficiency of the nuclear fusion reaction, and mass of the ship, and is a constant over time. In other words, the ship's velocity will increase linearly with time.

The limit to this velocity increase is the speed of light, and close to the speed of light the equation derived above will break down due to the effects of special relativity.

If we assume a threshold mass-collection rate *dm*/*dt*
(the units are mass per unit time), then the rate of mass collection
by the funnel
π *r*^{2} ρ *V* needs to be greater than the threshold.
This gives a threshold velocity:

Below this velocity, the ramjet engine will not work.V> (_{t}dm/dt) / ( πr^{2}ρ )

In order to get up to the threshold velocity, a ramjet may be equipped with a reaction engine with its own power and reaction mass supply. This engine can be switched off once the ramjet begins to work.

Consider a ramjet moving at speed *V* with respect to the
interstellar medium. If matter collected by the funnel is stored in
the ship, then in a time increment *dt* an amount of mass
*dm* is given a change in momentum equal to the change in momentum
of the ship, but in the opposite direction:

But the mass collected in this time interval is as given underdm V= -M dV

πIntegrating with respect tor^{2}ρV^{2}dt= -M dV

dt/dV= -M/ ( πr^{2}ρV^{2})

Rearranging to make speed the subject as a function of time:t= [M/ ( πr^{2}ρ ) ] (1 /V- 1 /V)_{o}

Note that the drag generated on the ship by the incoming interstellar medium does not affect the acceleration calculated above, since only the total change of momentum is relevant (and is how the acceleration was calculated).V=M V/ (_{o}M+ πr^{2}ρV)_{o}t