Applications of Calculus 2 to Physics

I'm teaching a section of Calculus 2 (integration techniques, arc length, surface area, improper integrals, parametric & polar functions, sequences, and series ) next semester and would like to assign a problem set near the end of the course to each student relevant to his or her major.

It can be expected that a handful of students will be majoring in physics. These students are going to be mostly first years, so they will have little exposure to physics outside what they learned in high school.

I have the lightest of physics backgrounds so I could use some advice in developing the problem set.

• What are some interesting applications of the material taught in Calculus 2 to physics that require little in the way of physics prerequisites?
• Are there any good resources I may want to look into for ideas?

Note: I asked this question on physics.SE but it was closed and users recommended I ask it on matheducators.SE.

• Distance, velocity, and acceleration are fundamental and a nice opportunity to use derivatives and integrals. – HopDavid May 30 '14 at 12:55
• Wow, this sounds like shooting fish in a barrel compared to most concerns of this type! I would simply flip through a lot of calculus texts (in a colleagues' office, in the library, etc.), maybe focusing mostly on texts published before the 1980s or so when physics applications started getting drummed out of texts in favor of applications in ecology, food preparation, sewage management, etc. Look at chapter sections titled work, fluid flow, center of mass, moments of inertia, radius of gyration, etc. – Dave L Renfro May 30 '14 at 15:20
• you might look at page 200 and 207 of supermath.info/OldschoolCalculusII.pdf – James S. Cook May 31 '14 at 19:08
• How much time do you want the students to approximately spend on the projects? – Wrzlprmft May 31 '14 at 20:07
• Physics and calculus are very linked. Kinematics especially is a very intuitive way to think about derivatives and integrals. As Renfro said, just look at older calc textbooks and they are full of physics word problems (even current ones often are). You should also, just look at Halliday and Resnick (physics text for calculus based physics.) – guest Apr 21 '18 at 14:23

In special relativity, we have $\gamma=(1-v^2)^{-1/2}$, where $v$ is the velocity in units of the speed of light. Relativistic momentum for a particle with $m\ne0$ is $p=m\gamma v$, and kinetic energy is $K=m(\gamma-1)$ (in units where $c=1$). (a) Expand $p(v)$ in a Taylor series and show that the lowest-order nonvanishing term recovers the nonrelativistic limit. (b) Do the same for $K$.

Polar coordinates can be used to calculate things like the moment of inertia of a disk.

The magnetic field of a long, straight wire is of the form $B\propto 1/r$. The energy density of the field (energy per unit volume) is proportional to $B^2$. Show that the improper integral diverges logarithmically at both $r\rightarrow0$ and $r\rightarrow\infty$. (Physically, the wire can't have zero radius, and the distant field isn't realistic because we need a complete circuit.)

For an object close to a concave mirror, the object's distance $u$ from the mirror and the image's distance $v$ from the mirror are related by $1/f=1/u-1/v$, where $f$ is a constant (the mirror's focal length). The magnification is $M=v/u$. What happens to the magnification in the limit as $u\rightarrow0$? (This gives an indeterminate form.)

If an atom of a radioactive isotope is observed not to have decayed at $t=0$, then the probability distribution for the time at which it will decay is of the form $ke^{-bt}$. Use normalization to relate $k$ to $b$.

What I try to emphasize is that any formula they're familiar with requires integral calculus once some of the quantities vary.

Basic example: (distance)=(rate)*(time). How far do you go if your rate is a function of time?

Medium example: (force)=(mass)*(acceleration). What is the end velocity of a rocket that burns mass (i.e. fuel) to exert constant force?

Advanced example: (Work)=(Force)*(distance). What if you have a cube of butter sliding on a hot pan? The force of friction is proportional to the mass, but the mass is decreasing because it's melting. The more you push it, the faster it melts (since the butter can spread away easier). So how much work is done in pushing a pad of butter around until it melts in your pancake griddle?

I would say that Taylor series come up fairly often in physics (both elementary and not):

• E.g., the force due to gravity is usually given by $F=mg$ but this is merely an approximation. See exercise 4 here for example.
• E.g., the motion of simple pendulum can be described via a Taylor series. See here.
• E.g., the first order approximation to the binomial $(1+x)^n$ has applications to electrostatics. See here.
• The $n$-body problem has a formulation which can be solved via Taylor series.
• Position given by $$x(t)=x_0+v_0(t-t_0)+a/2(t-t_0)^2$$ is really just a power series approximation for position.
• They also have various applications to Statistical Mechanics. See here

The "classic" Calculus 2 application to Physics are the Frenet formulas, which incorporate tangents, normals and binormals.

• my idea of calculus 2 is wrong, we only get to Frenet-frames in calculus 3. Even so, some instructors (sadness) omit discussion of curvature and the T,N,B frame. – James S. Cook May 31 '14 at 19:02
• In my "time," (the 1970s), I got this at the end of Calculus 2 (in high school). Then at the beginning of the following Calculus 3 class, in college. – Tom Au Jun 1 '14 at 14:03

I'd probably stay away (or use very simple examples) if I didn't really understand the physics behind the application. As several of the other answers show, many applications involve approximations or the simple models give formulas that fail due to being non-realistic in some way. You are asking how to explain use of a tool, if you don't know the application, you can't really use the tool on it.