Numbers, Symbols, and Physics

Numbers, Symbols, and Physics

All physics problems ask us to figure something out about the world that either we didn't know before, or we were not sure if it was true. Sometimes we'll want to use our physics tools (i.e. Vectors, $F=ma$, etc) to arrive at an actual number that reflects a physical quantity. This is a numerical calculation (or quantitative). Other times, we'll want to know a more general relation between two physical quantities. For example, how does air pressure depend on vertical elevation? We'll call this a symbolic calculation. The following attempts to show the difference between these when it comes to problem solving.

We'll start with a numerical calculation. How fast will I be going when I hit ground after falling from a 10 foot ladder. This would help me determine whether or not I wanted to fall off a 10 foot ladder. We can solve this using kinematics:

$$ \begin{eqnarray} v^2 = v_0^2 + 2 g y = 0 + 2 \times 9.8 \; \textrm{m/s}^2 \times 10 \; \textrm{ft} \times (.3048 \; \textrm{m/ft}) \\ v = 7.73 \; \textrm{m/s} \end{eqnarray} $$

That's pretty fast, so I probably don't want to fall off a 10 foot ladder. This is useful if all there were in the world were 10 foot tall ladders. Of course, there are many different height ladders. Should we redo this calculation for each? Not if we value our free time!

Instead, let's rephrase the problem as this: find the relation between speed of impact and ladder height. We'll derive the relation using basic kinematics:

The position of a freely falling body is given by:

$$ y = y0 + v{0y} t + \frac{1}{2}a_y t^2 $$

For this situation, the final position is 0, (i.e. the ground), the initial velocity is 0, and the acceleration is $g$ in the negative $y$ direction. Thus the equation reduces to

$$ 0 = y_0 - \frac{1}{2}gt^2 \Rightarrow y_0 = \frac{1}{2}gt^2$$

I can use this to get the time it take to fall from a height $y_0$

$$t = \sqrt{\frac{2 y_0}{g}}$$

Now the velocity of an accelerating object is given by:

$$v_y = a_y t = - g t$$

so, for an object falling, we expect the velocity to be:

$$v = - g \left( \sqrt{\frac{2 y_0}{g}} \right)$$

This simplifies to:

$$v = -\sqrt{2 g y_0}$$

Now, we have a relationship between velocity, little $g$ and the initial height. The problem has been solved in a general way. Using this relation, we can understand how the speed is related to height. Double the height for example, doesn't double our final speed. Interesting! This is an example of symbolic manipulation that leads to a algebraic relation. Usually, this will be a more powerful approach to solving a problem, especially as they get more complicated.

Things to note about symbolic manipulations:

Things to note about numerical calculations.

How do you know if you should do a symbolic or a quantitive approach? You'll need to look at the problem. If there are a bunch of numbers, then probably you'll be expected to get a number in the end. If its full of letters and things like $v_0$, it's probably a symbolic problem.

Another clue is whether it's even possible to reduce things to just a number. If not, then you'll leave it in a symbolic form. One skill you're acquiring while studying physics is the ability to determine what you can find solutions for and what you cannot.

For example, in the second example above, we could re-word it as: find the final speed of an object dropped from a height $h$ above the ground. Now, there is no way we can find a number for that. We would need a value for $h$ in order actually arrive at a numerical answer. So, we best leave it as letters: $v = - \sqrt{2 g h}$. If the problem said: find the final speed of an object dropped from a height of 10 meters, then sure, we can certainly find a number for that: 14 m/s .

In closing, both types of reasoning are important and necessary. To actually build a bridge or design a circuit, you'll need to be able to assign values to quantities. However, the more general symbolic approach will be useful when determining if it's even possible to build the bridge or circuit you have in mind.