The relation that satisfies the the dimensional analysis is [tex]1[LT^{-1} ]= k[L^{p+q} T^{-2p}][/tex]
Your velocity equation seems to be incorrectly typed, below is the equation is I presume you intended to type;
[tex]v = kg^{p}h^{q}[/tex]
The dimensional analysis of the velocity equation is calculated as follows;
[tex]v = kg^{p}h^{q}[/tex]
where;
k, p and q are constant
h is the distance (m)
g is acceleration due to gravity (m/s²)
v is the velocity (m/s)
[tex]v = kg^ph^q\\\\LT^{-1} = k[LT^{-2}]^p[L]^q\\\\LT^{-1} = k[L^pT^{-2p}][L^q]\\\\1[LT^{-1} ]= k[L^{p+q} T^{-2p}]\\\\1 = k\\\\L^1 = L^{p+q} \\\\1 = p+q\\\\T^{-1} = T^{-2p}\\\\[/tex]
Thus, the relation that satisfies the the dimensional analysis is [tex]1[LT^{-1} ]= k[L^{p+q} T^{-2p}][/tex]
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