Equations are the way we express relationship between physical quantities, represented by algebraic symbols. Each such symbol denotes a number and unit, for example, d may represent a distance of 10m, t a time of 5s, and v, a speed of 2(sm).
Equations must be dimensionally. consistent, meaning two terms can be added or equated only if they have the same units.
For instance, if some object moves at a constant speed v, and travels distance d in time t, we can relate these quantities with the following equation:
d=vt.
If the unit associated with d is meters, then the product v(t) must also be expressed in meters:
10m=(2sm)(5s).
Because the unit s in the denominator of sm is canceled out, the only unit left is m, which are the meters in which the results are expressed.
Problem Solving Strategy 1.2
Identify Relevant Concepts: In most situations, the SI units are what should be used to solve the problem, and if other units are needed, make the conversions to the solution of the problem, instead of attempting to do so mid-solution.
Set Up & Execution: Units are multiplied and divided in the same manner as regular algebraic symbols:
5k×2=10k,4v+4v=8v.
This allows for easier conversion between quantities expressed in one type of unit to another type of unit: express the same quantity in two different units and form an equality.
For instance, when we say 1min=60s, we're not equating the numbers 1 and 60, but rather that 1min represents the same time interval as 60s, meaning the ratio 60s1min=1, as does the reciprocal, 1min60s.
We can take a quantity by either factor, without changing the physical meaning of the quantity:
3min=(3min)(1min60s)=180s.
If you carry out the unit conversions correctly, the undesired units will cancel out.
Also make sure that the answer is reasonable: since the second is a smaller interval than a minute, the result of the calculation should contain a higher number of seconds than minutes in the same time interval.