Dimensional analysis with sums

Question

Lets say I am given

$$m \cdot \frac{dv}{dt} = mg - \beta v^2 $$

Where $m$ is mass, $g$ is gravity, $\beta$ is unknown units, $v$ is velocity.

Let

Length - $L$

Mass - $M$

Time - $T$

So using units we write as

$$M\cdot(\frac{L}{T^2}) = M\cdot \frac{L}{T^2} - [\beta] \cdot \frac{L^2}{T^2}$$

But then I get $[\beta] = 0$?

Answer 1

You can't add thing that don't have the same dimension so : $$[mg]=[\beta v^2]=M \frac{L}{T^2}$$ and $$[\beta]=\frac{M}{L}$$

Answer 2

You never subtract /add dimensions. Only dimensions should match for equations to be valid. So $[B]\frac {L^2}{T^2}=\frac {ML}{T^2}$. Thus $[B]=\frac {M}{L}$.