Video Transcript
What are the dimensions of a quantity that is equal to momentum divided by time?
Let’s say that the quantity whose dimensions we want to solve for is called 𝑄. This quantity is equal, we’re told, to a momentum 𝑃 divided by a time 𝑡. In general, dimensions of a quantity are different than the units of that quantity. For example, the units of time could be seconds, but the dimensions of time, whether that time is expressed in seconds or minutes or hours or some other unit, is capital 𝑇, time. Dimensions then are not units, but they are the quantities that units express.
Let’s now think for a moment about the momentum in the numerator of this fraction. Classically, the momentum 𝑃 of an object equals that object’s mass multiplied by its velocity. The dimension of mass 𝑚 is simply capital 𝑀. This tells us that this symbol here refers to mass. And then when it comes to the dimensions of velocity 𝑣, we can recall that the velocity of an object equals its displacement divided by time. A displacement is a length 𝐿, and time is a time. The dimensions of velocity are capital 𝐿 over capital 𝑇.
All this means that the overall dimensions of 𝑚 times 𝑣, the overall dimensions of momentum 𝑃, are mass times length divided by time. We know then the dimensions of our numerator 𝑃 and our denominator 𝑡. The dimensions of the quantity 𝑄 then are 𝑀 times 𝐿 divided by 𝑇 all divided by 𝑇. If we multiply numerator and denominator of this fraction by one divided by 𝑇, then 𝑇 cancels entirely in the denominator. And our result is 𝑀 times 𝐿 divided by 𝑇 squared. This is the same thing as 𝑀 times 𝐿 times 𝑇 to the negative two.
The dimensions of 𝑄 then, a quantity equal to a momentum divided by a time, are mass times length times time to the negative two.
