Question

How the following equations are dimensionally correct -

[F]=G Mm/r^2

And

V^2 - u^2=2as

Answer 1

For the first equation
F = GMm/r^2 ---------> eq.(1)
Dimensions :
F = [MLT^-2]
G = [M^-1 L^3 T^-2]
M = [M]
m = [M]
r^2 = [L^2]
Substituting the values in equation 1

[MLT^-2] = [M^-1L^3T^-2] x [M] x [M] / [L^2]

[MLT^-2] = [M]^-1+2 x [L]^3-2 x [T]^-2

[MLT^-2] = [M]^1 x [L]^1 x [T]^-2

[MLT^-2] = [MLT^-2]

L.H.S = R.H.S
Therefore Newton's law of gravitation is dimensionally correct.

for the second equation

V= final velocity = [LT-¹]
u= initial velocity=[LT-¹]
a= acceleration =[LT-²]
s= distance= [L]
v²=u²+ 2as
=>v²-u²=2as
[LT-¹] ²- [LT-¹]²=2[LT-²][L]

[L²T-²][L²T-²]=2[L²T-²]

2[L²T-²]=2[L²T-²]

L.H.S=R.H.S
hence proved
Therefore the given equation is is dimensinaly correct.