Video Transcript
What is the name of the missing unit U in the formula speed in meters per second is equal to distance in U divided by time in seconds?
The physical relationship we are working with is the definition that speed is equal to the distance traveled divided by the time that it took to travel that distance. Since this is a physical formula, the units on both sides must agree. That is, the units of speed must be the same as the units of distance divided by time. Note that it doesn’t matter what the units are as long as they agree on both sides of the equation. So, whether we measure speed in kilometers per hour or centimeters per minute, as long as the units agree, we have a valid equation.
Now, in this question in particular, we are told that the units we should use for speed are meters per second, represented by the symbol m divided by s. We are also told that the unit for time is seconds, represented by the symbol s. And finally, the units for distance are the unknown units U. What we see is that meters per second must be the same units as a quantity where the numerator has units of U and the denominator has units of s.
We now recall that units behave like mathematical variables. That is, to find the units when dividing two quantities, we simply divide the individual units. So the units of the right side of our expression are U divided by s or U per second. Therefore, because the two sides have to have the same units, U per second is the same as meters per second. Since both of these units are something per second, the only way for U per second to be the same as meters per second is if U is identically meters.
This is actually not at all surprising if we recall that meters are a unit of distance and seconds are a unit of time. It is therefore entirely expected that the units for distance and time in the quantity distance per time would be exactly the same as the units for distance and time that make up the unit for speed.
