Physics uses symbols to represent physical quantities that describe some aspect of the real world. For example, the physical quantity time is typically represented by the letter \(t\text{.}\) Symbols are extremely versatile: the letter \(t\) is used to represent any specific instant in time, such as \(t = 3\) s, or a generic, unspecified instant, time \(t\text{.}\) Different times are further specified by the use of subscripts, such as \(t_i\) for initial time, \(t_f\) for final time, or \(t_4\) for the fourth in a series of related times. Every symbol also carries its own appropriate units! For example, you would not want to write “I turned on the machine at \(t\) seconds”; instead, you would write “I turned on the machine at time \(t\text{.}\)”
Key Skill3.7.1.Solving a Symbolic Equation.
From an equation in which only one quantity is unknown, you should be to solve the equation for that unknown quantity (when possible) without plugging in numbers. The primary reasons for solving equations symbolically rather than numerically are:
It is easier to avoid and to catch algebra errors when solving symbolically, because common errors can be traced backwards through your algebra
There are many situations where you do not have any numerical quantities
When you solve an equation symbolically, you have solved it for all possible values of the other quantities, not just the particular numbers you have for a given situation
Symbolic equations allow you to do powerful sensemaking above and beyond what is possible with numerical quantities alone
Physics uses symbolic equations to represent quantitative models of the real world. An example equation is shown below.
\begin{equation*}
\Delta x = v_{x,ave}\Delta t
\end{equation*}
In this equation, t represents time, x represents the position of an object along the x-axis, and \(v_{x,ave}\) represents the average velocity of the object in the x-direction. The symbol \(\Delta\) transforms both \(x\) and \(t\) to specify change in position and change in time, respectively.
Suppose you know both \(\Delta x\) and \(v_{x,ave}\text{;}\) determine an equation for the amount of time required to travel this distance.
Since \(\Delta t\) is multiplied by \(v_{x,ave}\) on the right hand side of the equation, you can divide both sides by \(v_{x,ave}\) to get \(\Delta t\) alone!
\begin{equation*}
\Delta x = v_{x,ave}\Delta t
\end{equation*}
\begin{equation*}
\frac{\Delta x}{v_{x,ave}} = \Delta t
\end{equation*}
\begin{equation*}
\Delta t = \frac{\Delta x}{v_{x,ave}}
\end{equation*}
Physics is full of symbolic equations, and one context might often have several known and unknown quantities and many symbolic equations that relate those quantities. Thus, you will often be asked not only to solve symbolic equations, but to solve systems of symbolic equations.
Key Skill3.7.3.Solving a System of Symbolic Equations.
Solving a system of symbolic equations is no different from solving any other system of equations: you just might not have any numerical quantities. Below are a few things to keep in mind:
You need a number of independent equations equal to the number of unknown quantities; remember that known quantities, even though they are still written as variables, do not count
If you have too many equations, it may be that some of them are not independent, or it may be that some are not relevant to the physical situation
If you have too few equations, you usually need more information to make progress: try looking for another equation relevant to the physical situation
Simple substitution is usually sufficient to solve most systems of equations in physics
If you have multiple unknown quantities but you only need to find one of them, solve for the other unknown quantities first when using substitution. It may seem counterintuitive, but you want the last equation you plug everything into to have only the unknown quantity you care about in it!
From elementary chemistry, the pressure \(p\text{,}\) volume \(V\text{,}\) and temperature \(T\) of an ideal gas undergoing adiabatic expansion may be related by the following pair of equations, in which \(n\text{,}\)\(R\text{,}\)\(\gamma\text{,}\) and \(\alpha\) are positive constants:
Suppose you want to know the exact temperature of the gas at the end of the expansion. There are only two equations and three unknowns, so it is necessary to specify a temperature: suppose you know the volume at the end of the expansion is \(V_f\text{.}\) Since you want to know temperature, you want to solve one of the equations for the other unknown, pressure, then substitute that expression for pressuring into the other equation. The second equation looks a little more complicated, so start there.
There is one special exception to solving equations without plugging in numbers: the number zero. When a quantity is known to be zero, it is often the result of some special, fundamental property of the situation; thus, you are unlikely to want to go back and change the number away from zero (if you do, you often make the situation more complicated). So if you have a quantity you know is zero, feel free to plug it into any equations early!
ExercisesPractice Activities
Figure3.7.6.A velocity vector and its \(x\)-component.
Activity3.7.1.Velocity Manipulation.
The diagram above shows a velocity vector and its \(x\)-component, \(v_x\text{.}\)
(a)Vector Components.
Write a symbolic equation that gives \(v_x\) in terms of \(v\) (the magnitude of \(\vec{v}\)) and the given angle \(\theta\text{.}\)
You have an air cart with mass \(m\) that is subject to two constant forces: thrust and friction. You conduct two experiments with the air cart. In Experiment 1, the thrust force points forward and the friction force points backward, leading to a forward acceleration of \(a_1\text{.}\) In Experiment 2, the thrust and friction forces both point backward, leading to a backward acceleration of \(a_2\text{.}\)
(a)Unknown Quantities.
Identify the unknown quantities of interest and give them an appropriate symbol.
