Question

Check the dimensional consistency of the following equations :
(i) `upsilon = u +at` (ii) `s = ut +(1)/(2) at^2`
(iii) `upsilon^2 - u^2 = 2as`

Answer 1

(i)This equation gives the relation for average velocity , `v_"av"` checking the dimensions on both sides.
Average velocity `[V_"av"]=[LT^(-1)]to` L.H.S.
Initial velocity `[u]=[LT^(-1)] to` R.H.S.
Final velocity `[v]=[LT^(-1)]`
Since all the terms have the same dimensions , the equation is dimensionally consistent . But it is not correct physically. The correct expression for average velocity for the given case is `v_"av"="u+v"/2`
(ii)`s=ut+1/2at^2`
[s]=[L] `to` L.H.S.
[ut]=[u][t]
`=[LT^(-1)][T]`
[ut]=[L] `to` R.H.S.
`[1/2at^2]=[LT^(-2)][T^2]=[L] to` R.H.S.
Each term of the given equation has the same dimensions, namely that of length . Hence, the equation is dimensionally consistent. It is correct physically too.
(iii)`v^2-u^2="2s"/a`
Checking the dimensions on both sides .
`[v^2]=[LT^(-1)][LT^(-1)]`
`=[L^(-2)T^(-2)] to `L.H.S.
`[u^2]=[LT^(-1)][LT^(-1)]`
`=[L^2T^(-2)]`
`["2s"/a]=[L]/[LT^(-1)]=[T] to` R.H.S.
Dimensions of the quantities on both sides are not same. So the equation is not correct dimensionally, and hence , physically it cannot be correct.