# Is the equation $Snth =u+(1/2) (a)(2n−1)$, which is used to find displacement in $n$th second dimensionally correct?

In physics we come across expressions like ° C = 5/9 (° F - 32)

K = ° C + 273 Used for temperature conversion from one scale to other. Also we come across expressions like Velocity of a body as a function of time is $$v=3t^2$$. Or the displacement travelled in nth second in case of uniform acceleration is given by the expression: $$Snth =u+(1/2) (a)(2n−1).$$ Do all they need to be dimensionally correct or they are just numerical equations and not generalized physical equations which need not to be dimensionally correct? I had searched the same on net but did not find any satisfactory answers. As some sources say in displacement equation Snth t is hidden in some terms but how can something be hidden in a generalized physical equation.

• unless you define the symbols and their units it is not possible to debate anything regarding combinations of these units. Jan 15 at 15:09

I will provide you the algorithm:

Suppose we have an equation/formula : A = B + CD/2

Here A, B, C, D are physical quantities with appropriate dimensions. Now if A should be dimensionally correct then the terms B and CD/2 should have the same dimensions as of A.

A few things you should keep in mind while using dimensional analysis, which are as follows:

1. Sometimes more than 1 physical quantity have same dimensional formula. Ex: Workdone and Energy have the same dimensional formula. Torque and Angular Momentum also have same dimensional formula.

2. Dimensional analysis is not the most profound way of checking correctness of equation. Ex: Newton's equation of motion sound good dimensionally but don't possess symmetry accordance to Lorentz Transformation. On the other hand Maxwell's equation are correct dimensionally as well as possess symmetry.

So, a dimensionally right formula may or may not be symmetric but a dimensionally wrong equation will never possess symmetry.

Hope this helps.

Assuming "u" is a speed, "a" an acceleration, and "n" a count of units of time, then the right side of your equation has "dimensions" of length over time, but a displacement on the left side is a length. The equation is not correct unless you assume that the left side is divided by a unit of time.