# I'm having trouble understanding the dimensions of this falling body problem

I'm working with this:

A 50 kg mass is shot from a cannon straight up with an initial velocity of 10m/s off a bridge that is 100 meters above the ground. If air resistance is given by 5v determine the velocity of the mass when it hits the ground.

I haven't much experience with forces, but I understand that their dimension is $\frac{mass\times length}{time^2}$.

So if I have some force acting on a mass, then these should be the dimensions. The force acting on the mass in the question from gravity is 50(9.8)N, which has the correct dimensions. The next step in the solution tells me that $mv'=mg-5v$.

I don't understand this. $v$ has dimensions $\frac{m}{s}$. Am I supposed to assume that 5 has dimensions $\frac{mass}{sec}$? If so, what kind of a unit is this?

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## migrated from math.stackexchange.comNov 18 '12 at 22:01

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you should assume that 5 has dimensions of $\frac{Ns}{m}$. – Tpofofn Nov 18 '12 at 4:04
Anyway, yes, the $5$ has dimensions of force per unit velocity = $\mathrm{N\:s\:m^{-1} = kg\:s^{-1}}$. This is no more unusual than if I were to say a thermometer's mercury level rises 2 millimeters whenever the temperature goes up by a degree Celsius, giving units of $\mathrm{m\:K^{-1}}$. – Rahul Nov 18 '12 at 19:42