Calculus and mechanics are often taught as separate subjects. It shouldn’t be like that. If you learn calculus without mechanics, it will be boring. If you learn physics without calculus, you won’t truly understand.
I think I may have found a way to solve this chicken and egg problem. It goes a little something like this:
- Chapter 1. You need [solving_equations,algebra,quadratic_equation] to do physics. That is all the prerequisites for first year Physics.
- Chapter 2. Physics laws are expressed as equations. If you know how to solve equations, then you know how to solve physics equations. In particular we will study the kinematics equations $x(t)$, $v(t)$, $a(t)$, which describe the motion of an object.
- Start by defining kinematics concepts like time $t$, position $x(t)$, velocity $v(t)$, acceleration $a(t)$, initial position $x_i$, and initial velocity $v_i$. We can then state the UAM equations straight up: $a(t)=a$, $v(t)=v_i+at$, $x(t)=x_i+v_it+\frac{1}{2}at^2$. Example (free fall): An object on which only the force of gravity acts is said to be in free fall. Such objects experience a constant downwards acceleration of magnitude 9.81[$m/s^2$]. The classic examples are a ball thrown in the air. Using equations you can calculate the trajectory of the ball, and predict where it will land. Equations are cool and all, but where do these equations come from?In order to find out we must take a short excursion into calculus-land.
- Calculus is the study of functions. We use calculus in to describe how quantities change over time (derivatives \(f'(t)\)) or to find the total amount of quantities that vary over time (integration \(F = \int f \;dt\)). Integrals sound fancy, but are really just a an area-under-the-curve calculation. Provide visual proofs for two important cases: if $f(t)=3$, then $F(t)=3t$. If $g(t)=t$, the integral is $G(t)=\frac{1}{2}t^2$.But why should anyone care about integrals? What good is computing the area under a curve?
- Integrals are the inverse operation of the derivative. In analogy with the inverse functions that we use when solving equations, the concept of an inverse operation is a useful concept in calculus: integrals are the inverse operations of derivatives.The kinematics equations of that describe the motion of objects can be derived from Newton’s law $F=ma$ and applying the integration operation twice. This is easy to see: we start by rewriting $F=ma(t)$ as $F=mx”(t)$, which means the force on an object is equal to the second derivative of $x(t)$. Recall that we just learned that integrals are the inverse operation of derivatives, so if we want to solve for $x(t)$ in $F=mx”(t)$ we can do it! First we divide both sides by m, in order to isolate the x expression on the right $F/m = x”(t)$. Then apply the integration operation twice in order to undo the two derivative operations.In particular, let us consider the case when $F=\textrm{const.}$ which implies that then $a(t)=\textrm{const.}=a$. The equation we want to solve is $F/m = a=x”(t)$. Applying the integration operation to both sides of this equation we get $at+C=x'(t)$. By definition $x'(t)=v(t)$ so the constant $C$ can be identified as the initial velocity $v(0)=v_i$. Applying the integration operation to both sides a second time gives us $\frac{1}{2}at^2 + v_it + x_i = x(t)$. This is how the UAM equations are derived: $F=ma$ and 2x integration steps.
- Main idea of this book: understand the math + physics is easier than just learning physics by memorizing the equations. With memorization, you would need to remember three equations of motion as separate entities. If you understand derivatives and integrals then you can remember just one equation $a(t)=a$, which is not much to remember since it is in the name UAM.
- We have now seen kinematics in one dimension. But the real world is three dimensional so we need to learn about the math for dealing with objects in 3D.
- Chapter 3: Vectors.
- Chapter 4: Now that we know about vectors we can discuss more physics (mechanics).
- Projectile motion. The position of the object is now a vector $\vec{r}(t)=[x(t),y(t)]$. There are two separate sets of kinematics equations. $x(t)$ is UVM (since no forces in the hz direction) while y(t) is UAM ($a_y=-9.81$ due to the force of gravity).
- Introduce dynamics $\vec{F}=m\vec{a}$, i.e. forces cause acceleration. Forces. Force diagrams.
- Momentum.
- Energy.
- Uniform circular motion.
- Angular motion.
- SHM.
The structure in Chapter 2 is the only new thing. After that, Chapter 4 is pretty much a standard course through the mechanics curriculum. So how is Chapter 2 so special, as to be worth blogging about at 2:44 in the morning?
I will tell you in point form, because it is kind of late indeed:
- It connects nicely with the Precalculus chapter. You just learned how to solve equations for 50 pages, and now I am telling you that you can do physics with this equation solving skill. Yey! Math is useful.
- Then we introduce a bit of basic kinematics concepts $x$, $v$, $a$ and the equations of motion. But then we say where did these equations come from (this is kind of a weak point?). To tell you, we must learn Calculus.
- Bam—drette là—we do a mini course on calculus in 5 pages. Integrals with pictures and FTC. Sure it is complicated but the analogy to f and f-inverse should make it go through.
- Then show derivation of $x(t)$ via int( int($F/m$) ). I can use the exact integral formula since students just saw those formulas as pictures 4 pages ago so they can’t say “i don’t know integrals”.
- Having this early exposure to integrals also helps with the work and potential energy section later on in Chapter 4.
- Basically the 5-page mini introduction to integral calculus is sufficient to do calculus-based mechanics course. The sin/cos derivative info and the chain rule required for deriving the SHM is presented in a just-in-time manner (i.e. in the last chapter).
But who am-I to say what is and isn’t a good way to teach. Only the students can tell me.
Update May 2021: I improved the formatting of the equations in the blog post and added links to the current version of the MATH & PHYS book preview.
