Question

The distance of a particle is x=a+bt+ct²+dt² where t is time. find the dimension of a, b, c and d please give the answer quickly

Answer 1

Like quantities or Quantities of same nature can be added or subtracted....For example , 5 meter can be added to 2 meter as both are distances..and are like quantitues or Quantities of same nature...

But 5 meter cannot be added to 5 minutes as quantities are of different nature...

So if X is a distance of a particle , then each terms on the RHS ( Right Hand Side ) must be distance also....
( And sum of individual terms on the RHS is also a
distance of a particle )

so Here
a = distance = SI unit is meter => Dimension = [L]

bt = distance = SI unit is meter => Dimension = [L]
=> Dimension of bt = [L]
=> b x t = [L]
=> b = [L] / t ( both sides dividing by t )
=> b = [L] / [T] ( Dimension of time = [T] )
so , Dimension of b = [L] x [T^-1 ] = [ LT^-1]

ct^2 = distance = SI unit is meter => Dimension = [L]
Dimension of ct^2 = [L]
c x t^2 = [L]
c = [L] / t ^2 ( on divinding both sides by t^2 means t squared)
c = [L] / [ T^2] = [ L T^-2]
So, Dimension of c = [ LT^-2]

Similary , we can find the dimension of dt^2 by doing the same process as we did above to find the dimesion of ct^2....

So, Dimension of d = [L T^-2]