# College Math Teaching

## July 1, 2013

### Mathematics: aids the conceptual understanding of elementary physics

Filed under: applications of calculus, editorial, elementary mathematics, pedagogy, physics — collegemathteaching @ 4:52 pm

I was blogging about the topic of how “classroom knowledge” turns into “walking around knowledge” and came across an “elementary physics misconceptions” webpage at the University of Montana. It is fun, but it helped me realize how easy things can be when one thinks mathematically.

This becomes very easy if one does a bit of mathematics. Let $m$ represent the mass of the object; $F = 10 = ma$ implies that $a = \frac{10}{m}$ which isn’t that important; we’ll just use $a$. Now putting into vector form we have $\vec{a}(t) = a \vec{i}, \vec{v}(0) = V_i \vec{j}, \vec{s}(0) = \vec{0}$. By elementary integration, obtain $\vec{v} = at \vec{i} + V_i \vec{j}$ and integrate again to obtain $\vec{s}(t) = \frac{1}{2}at^2\vec{i}+(V_i)t\vec{j}$ which has parametric equations $x(t) = \frac{a}{2}t^2, y(t) = V_i t$ which has a “sideways parabola” as a graph.

Let’s look at another example:

So what is going on? Force $F = \frac{d}{dt}(mv) = \frac{dm}{dt}v + m\frac{dv}{dt} = 0$. The first term is thrust and is against the direction of acceleration. So we have:$1000 = m\frac{dv}{dt}$ which, upon integration, implies that $\frac{1000}{m} t + v_0 = v(t)$ and so we see that the rocket continues to speed up at a constant acceleration.

These problems are easier with mathematics, aren’t they? 🙂