# The dimensional formula of distance travelled in $n$th second

I read that the dimensional formula of distance traveled in $n^{th}$ second is same as that of velocity. Okay, the formula for the distance traveled in $n^{th}$ second is $s_t= u+\frac{a}{2}(2t-1)$ where $u$ is initial velocity, $a$ is uniform acceleration and $t$ is the time. If we proceed and expand the aforementioned formula we will get $s_t = u + at-\frac{a}{2}$. Now, the last term i.e. $-\frac{a}{2}$ is not a velocity and isn't the principle of homogeneity violated?

• My translation of Mr. Rennie's comment: An equation MUST be dimensionally consistent across the equal sign if it is to have any chance of being correct. Units MATTER in physics, and this is one area where I consistently have difficulty getting the concept across in my AP physics classes (I teach high school physics). If the equation is dimensionally inconsistent, it is JUST PLAIN INCORRECT! – David White Aug 3 '15 at 23:16

The distance travelled in a time $t$ is:

$$s = ut + \tfrac{1}{2}at^2$$

So the distance travelled between $t$ and $t - \Delta t$ is:

\begin{align} \Delta s &= s(t) - s(t - \Delta t) \\ &= ut + \tfrac{1}{2}at^2 - u(t - \Delta t) - \tfrac{1}{2}a(t - \Delta t)^2 \\ &= u\Delta t + \tfrac{1}{2}a(2t\Delta t - \Delta t^2) \end{align}

The equation you cite is obtained by setting $\Delta t = 1$, but remember that you're setting $\Delta t$ equal to one second not the dimensionless quantity $1$. So your equation should really be:

$$\Delta s = u\cdot (1 \space\text{second}) + \tfrac{1}{2}a(2t(1 \space\text{second}) - (1 \space\text{second})^2)$$

or multiplying this out:

$$\Delta s = u\cdot (1 \space\text{second}) + at(1 \space\text{second}) - \tfrac{1}{2}a(1 \space\text{second})^2$$

So it is dimensionally consistent.

It is dimensionally correct...

In the derivation, you take ut-u(t-1) which makes it appear as u, but is actually u*1 second...