  The typical velocity of a heavy quark in a quarkonium is a widely used
quantity, in this paper, based on the relativistic Bethe-Salpeter equation
method, we calculate the average values ${\overline{|\boldsymbol{q}|^n}}$ and $
\overline{|\boldsymbol{v}|^n}\equiv v^n$ of a heavy quark in a $S$ wave or $P$
wave quarkonium rest frame, where $\boldsymbol{q}$ and $\boldsymbol{v}$ are the
three dimensional momentum and velocity, $n=1,2,3,4$. For a charm quark in
$J/\psi$, we obtained $v_{J/\psi}=0.46$, $v^2_{J/\psi}=0.26$,
$v^3_{J/\psi}=0.18$, and $v^4_{J/\psi}=0.14$, for a bottom quark in
$\Upsilon(1S)$, $v_{\Upsilon(1S)}=0.24$, $v^2_{\Upsilon(1S)}=0.072$,
$v^3_{\Upsilon(1S)}=0.025$, and $v^4_{\Upsilon(1S)}=0.010$. The values indicate
that ${v^n} >{v^{n_1}}\cdot{v^{n_2}}$, where $n_1+n_2=n$, which is correct for
all the charmonia and bottomonia. Our results also show the poor convergence if
we make the {speed} expansion in charmonium system, but good for bottomonium.
Based on the $v^n$ values and the following obtained relations $v^n_{4S} >
v^n_{3S}> v^n_{2S}>v^n_{1S}$, $v^n_{4P} > v^n_{3P}> v^n_{2P}>v^n_{1P}$ and
$v^n_{mP}>v^n_{mS}$ ($n,m=1,2,3,4$), we conclude that highly excited quarkonia
have larger relativistic corrections than those of the corresponding low
excited and ground states, and there are large relativistic corrections in
charmonium system.
