Converting displacement to correct units

Question

I know that the equation for displacement is:

S = V₀ t + $\frac{1}{2}$at$^{2}$

With S being displacement in metres, t is the time is seconds, V₀ is the initial velocity in ms$^{-1}$ and a is the acceleration in
ms$^{-2}$.

However, if the time used in the equation is milliseconds then what would be the unit of the displacement S?

Answer 1

In SI you have:

$$S [m] = V_o\left[\frac{m}{s}\right]\cdot t[s] + \frac{1}{2} a \left[\frac{m}{s^2}\right]\cdot \left(t [s]\right)^2 $$

If you only used ms for time then 1 s = 1000 ms. So if the time was expressed in ms then 1 ms would be equal to $\frac{1}{1000}[s]=0.001 [s]$. Therefore:

$$S [m] = V_o\left[\frac{m}{s}\right]\cdot \frac{t}{1000}[ms] + \frac{1}{2} a \left[\frac{m}{s^2}\right]\cdot \left(\frac{t}{1000} [ms]\right)^2 $$

$$S [m] = V_o\left[\frac{m}{s}\right]\cdot \frac{t}{1000}[ms] + \frac{1}{2\cdot 10^6} a \left[\frac{m}{s^2}\right]\cdot \left(t[ms]\right)^2 $$

Answer 2

Just multiply the units:

S = V₀ t + $\frac{1}{2}$at$^{2}$

units = $ \frac m {ms} + \frac m {ms^2} ms = \frac m {ms} + \frac m {ms} = \frac m {ms} $

Answer 3

The units of S (as was asked in the query) is controlled by what distance units are used in the velocity and acceleration (feet, meter, miles, etc.); not the time. Simply use one millisecond as .001 seconds and all will come out right. For example, for 5 millisecond time plug .005 for time into the equation.