Calculus for physics students who haven’t finished calculus yet. Six activities, one idea: a derivative is a slope and an integral is an area. Shrink a secant line until it becomes a tangent, add up rectangles until they become an integral, and see why a falling hamster with a parachute stops speeding up.
The six activities
- Secant → Tangent. Penguin Percy slides down an icy ramp. Shrink Δt, record the average velocity each time, and watch it settle on the instantaneous velocity.
- x → v → a. Three linked graphs with a time cursor and tangent lines. In Predict mode the velocity and acceleration graphs are hidden, and scrubbing leaves a trail of slope dots until you reveal them.
- The Area Machine. Left, right, middle or trapezoid rectangles under a curve, compared with the exact answer from the Fundamental Theorem of Calculus. Area below the axis counts negative, just like moving backward.
- Rule Drill. Endless derivative and integral practice: power rule, sine and cosine, e and ln, chain rule. Answers are checked numerically, so any correct way of writing it counts.
- Drag Lab. Hank the Hamster parachutes with linear or quadratic air resistance. The exact solution from separation of variables is drawn next to a step-by-step simulation that only uses Newton’s second law, along with terminal velocity and the time constant.
- Slice-a-Rod. Cut a rod with changing density into N slices of mass dm. Watch the sums for mass, center of mass and rotational inertia converge to the integrals as N grows, then move the rotation axis.
For teachers
Most of the integrals in AP Physics C: Mechanics show up in the fall, but students in AP Calculus AB usually don’t reach integrals until winter, and separable differential equations come later still. This lab is the companion to a Calculus for AP Physics C student packet and teacher guide built for that gap: derivatives, integrals, and the moves only physics asks for, such as slicing an object into dm and solving the drag equation. It runs entirely in the browser, with no accounts and no data collection, and works on phones. If you’d like the packet and answer key, get in touch.
Direct link: brandoncomella-afk.github.io/physics-sims/calculus-lab.html
