Question

find the dimension of kinetic energy &potential energy with their unit

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Answer 1

Hello Dear 
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Lets see ....
we know Kinetic energy (K.E.) = 1/2 mv²
 Remember constant have no dimensions 
it is in the form of [M L T ]
so dimension of mass = [M¹ L° T°]
 dimension of velocity = [M° L¹ T⁻¹]²
Dimension of kinetic energy =  [M¹ L° T°][M° L¹ T⁻¹]²
                                             =  [M¹ L° T°][M° L² T⁻²]
                                             =  [M¹ L² T⁻²]
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Unit of KE = PE =  kg m² s⁻² or joule 
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Similarly Dimensions of Potential energy are :
                                             = [M¹][L¹T⁻²][L¹]
                                             = [M¹L²T⁻²]
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Answer 2

The dimension of any energy and work is same.
The dimension of energy is [M L^2 T^-2]
for e .g. -
potential energy:-
formula : m .g. h
dimension : [M] [L T^-1] [L] = [ML^2T^-2]
unit: jule , erg

kinetic energy :-
formula : 1/2 m.v²
dimension : [M] [LT^-1]² = [ML^2T^-2]
unit : Jule , erg.

note: the constant are neglected just like π,1/2,23..etc. in the dimension.