Everything we do in physics is about describing the physical world; every number and letter that you write represents some real, physical quantity. There are two main ramifications of this:

- There are restrictions to the calculations that we can do.
- We need a common way of making and reporting measurements so we can effectively and consistently communicate with each other.

There are many kinds of measurements that we can make on a physical system. The three that we will be working with the most in this book are length, mass, and time. We will work with other quantities (such as energy and force), but they can ultimately all be described in terms of length, mass, and time. Length and time are intuitive enough that I will not define them here. Mass is a bit more nuanced.

A decent high school level description of mass is the amount of “stuff” in an object. Note that this is different from the space an object takes up (it’s volume); you can take something soft and compress it down to a smaller size, and you have not changed how much “stuff” is there. Mass is also different that how much an object weighs; weight is a measure of how hard gravity pulls on an object, and the strength of gravity changes from place to place. You can weigh the same object at sea level and on top of a mountain, and find different results, even though the amount of “stuff” in your object has not changed.

However, referring to “the amount of stuff” is not precise enough for good scientific discussion. Mass is best thought of as a *measurement of how much an object resists changes in motion*. We’ll develop this description of mass more in chapter 6.

## 3.1 Physically meaningful calculations

It is meaningless to add or subtract values that represent different physical quantities. For example, you cannot add 42 miles + 10 hours. Sure, you can add the numbers (42 + 10 = 52), but there is no physical meaning to this.

You can multiply and divide values that represent different quantities, however. For example

\[

\frac{42\textrm{ miles}}{10\textrm{ hours}} = 4.2\textrm{ miles/hour}

\]

which tells you a speed: in one hour, you travel 4.2 miles.

Remember, in physics class *every* number and symbol represents some measurable physical quantity. We need to follow these rules for everything we do mathematically, including calculations of numbers and algebraic manipulations of equations.

#### Example 3.1

- 21 cm + 3.0 cm = 24 cm
- 21 cm + 3.0 s cannot be added. You cannot add centimeters and seconds together because they measure different quantities.
- (21 cm)(3.0 s) = 63 cm⋅s
- \(\frac{21\textrm{ cm}}{3.0\textrm{ s}}\) = 7.0 cm/s