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.

Example.

Screen shot 2013-07-01 at 11.25.49 AM

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:

Screen shot 2013-07-01 at 11.40.26 AM

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? 🙂

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2 Comments »

  1. yay for math!

    Comment by Edward — July 2, 2013 @ 1:13 am

  2. […] The bottom line: science is hard and some scientific facts, even non-quantum physics facts, are difficult to impossible to grasp without a command of mathematics (e. g. angular momentum) I’ve linked to other examples here. […]

    Pingback by The open mind, the so-called political center and the nonsense…. « blueollie — December 6, 2013 @ 3:47 am


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