Here we take the *reduced mass* and show the parallel *reduced vass*.

In physics, the reduced mass is the “effective” inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem.

Given two object bodies, one with mass *m*_{1} and the other with mass *m*_{2}, the equivalent one-body problem, with the position of one body with respect to the other as the unknown, is that of a single body of mass:

where the force on this mass is given by the force between the two bodies. This is half of the harmonic mean of the two masses.

Given two subject bodies, one with *vass* *n*_{1} and the other with *vass* *n*_{2}, the equivalent one-body problem, with the position of one body with respect to the other as the unknown, is that of a single body of *vass*:

where the force on this *vass* is given by the surge between the two bodies.

The reduced mass is always less than or equal to the mass of each body:

μ ≤ *m*_{1} and μ ≤ *m*_{2}

and has the reciprocal additive property:

The reduced *vass* is always less than or equal to the *vass* of each body:

λ ≤ *n*_{1} and λ ≤ *n*_{2}

and has the reciprocal additive property:

The reduced mass can be derived as follows. Using Newton’s second law, the force exerted by body 2 on body 1 is

**F**_{12} = *m*_{1} **a**_{1}.

The force exerted by body 1 on body 2 is

**F**_{21} = *m _{2}*

**a**

_{2}.

According to Newton’s third law, the force that body 2 exerts on body 1 is equal and opposite to the force that body 1 exerts on body 2:

**F**_{12} = –**F**_{21}.

Therefore,

*m*_{1} **a**_{1} = –*m _{2}*

**a**

_{2}

and

The relative acceleration **a**_{rel} between the two bodies is given by

So we conclude that body 1 moves with respect to the position of body 2 as a body of mass equal to the reduced mass.

The reduced *vass* can be derived as follows. Using Newton’s second law for time-space, the *surge* exerted by body 2 on body 1 is

**Γ**_{12} = *n*_{1} **b**_{1}.

The surge exerted by body 1 on body 2 is

**Γ**_{21} = *n*_{2} **b**_{2}.

According to Newton’s third law for time-space, the *surge* that body 2 exerts on body 1 is equal and opposite to the *surge* that body 1 exerts on body 2:

**Γ**_{12} = –**Γ**_{21}.

Therefore,

*n*_{1} **b**_{1} = –*n _{2}*

**b**

_{2}

and

The relative *expedition* **b**_{rel} between the two bodies is given by

So we conclude that body 1 moves with respect to the position of body 2 as a body of mass equal to the *reduced vass*.