Capstone: A Real Worked Mechanical/Thermal System
Nine chapters have built two toolkits: one for motion, force, and gravity (Chapters 1–6), and one for heat, engines, and entropy (Chapters 7–9). This capstone runs a single, continuous, fully worked system through every one of them — a small steam-launched roller coaster car, from the boiler that powers its launch to the braking collision that finally stops it.
The System
A steam piston launches a 400 kg roller coaster car (with rider) along a straight 3 m launch track. The car then climbs into a 2.5 m-radius vertical loop, completes the circuit, and returns to ground level, where it couples with a stationary 600 kg braking car to bring the ride safely to rest. Every number below is carried forward, unmodified, from the step before it.
Before launch, superheated steam fills the 0.05 m³ piston chamber at T = 400 K, with n = 3 mol present. Using the ideal gas law to find the steam's pressure:
P = (3 × 8.314 × 400) / 0.05
P ≈ 199,540 Pa ≈ 200,000 Pa
As the steam expands and drives the piston outward, the boiler supplies Q = 45,000 J of heat during the power stroke. The piston has area A = 0.05 m² and travels d = 3 m, so the swept volume is ΔV = Ad = 0.15 m³. Treating the expansion as roughly constant-pressure, the work done by the steam is:
ΔU = Q − W = 45,000 − 30,000 = 15,000 J
15,000 J of heat remains in the steam as increased internal energy; the other 30,000 J becomes the mechanical work that launches the car.
Measured over a complete piston cycle (this power stroke plus the return stroke needed to reset it), the boiler-and-piston system converts heat into net usable work at a real, measured 20% overall efficiency. With steam entering at TH = 400 K and exhaust condensing at roughly TC = 300 K, the Carnot maximum for any engine operating between these two temperatures is:
The engine's real 20% efficiency sits honestly below this theoretical 25% ceiling — exactly as Chapter 8's own theory requires, since no real, imperfectly reversible engine can reach, let alone exceed, the Carnot limit.
The steam's pressure acts on the piston's full area to produce the launch force:
a = F/m = 10,000 / 400 = 25 m/s²
Using v² = u² + 2as over the 3 m stroke (u = 0):
v = √150 ≈ 12.25 m/s
The launch takes t = v/a = 12.25/25 ≈ 0.49 s.
Checking the mechanical work directly:
KE = ½mv² = ½ × 400 × 150 = 30,000 J
Power output during the launch: P = W/t = 30,000/0.49 ≈ 61,200 W, or about 61.2 kW.
The car enters the loop (radius r = 2.5 m) at ground level with its full 30,000 J of kinetic energy. At the top of the loop (height = 2r = 5 m), some of that energy has converted to gravitational potential energy:
KEtop = 30,000 − 19,620 = 10,380 J
vtop = √(2 × 10,380 / 400) ≈ 7.20 m/s
At the top of a loop, the car needs a minimum speed for gravity alone to supply the required centripetal force (mg = mv²min/r):
The car's actual speed at the top, 7.20 m/s, comfortably exceeds this 4.95 m/s minimum — confirming the loop is safely designed, with real margin to spare, before a single rider ever boards.
Having completed the frictionless loop, the car returns to ground level with its full original speed, v ≈ 12.25 m/s, and couples with a stationary 600 kg braking car — a perfectly inelastic collision:
400 × 12.25 = 1000 × vfinal
vfinal ≈ 4.90 m/s
Checking the kinetic energy: before coupling, KE ≈ 30,000 J; after, KE = ½(1000)(4.90²) ≈ 12,000 J. Roughly 18,000 J has vanished from the mechanical system entirely.
Every Chapter, In One System
| Step | Chapter | What It Contributed |
|---|---|---|
| 1 | 9 — Kinetic Theory of Gases | Ideal gas law gives the steam's real pressure, ~200,000 Pa |
| 2 | 7 — The Laws of Thermodynamics | First Law: ΔU = Q − W for the power stroke |
| 3 | 8 — Heat Engines & Entropy | Real efficiency (20%) checked against the Carnot ceiling (25%) |
| 4 | 2 — Newton's Three Laws | F = PA gives launch force; F = ma gives acceleration |
| 5 | 1 — Kinematics | SUVAT gives launch speed and launch time |
| 6 | 3 — Work, Energy & Power | W = Fd confirms KE; power output during launch |
| 7 | 6 — Gravity Revisited | Conservation of energy gives speed at the top of the loop |
| 8 | 5 — Circular Motion & Rotational Dynamics | Minimum loop speed confirms a safe design |
| 9 | 4 — Momentum & Collisions | Perfectly inelastic braking collision; energy lost to heat |
Hands-On Exercises
Course Complete
Classical Mechanics & Thermodynamics is now complete, 10/10 chapters — from Chapter 1's own real, corrected Galileo tower story through this capstone's full mechanical-and-thermal system. Along the way, this course found a genuine recurring pattern: the popular, tidy version of a scientific story (Galileo's tower, "Newton's cradle," Newton's apple, the Zeroth Law's own name) is consistently less accurate, and less interesting, than the messier real history behind it.
Electromagnetism & Relativity (electromag1) is the second half of the original "Classical Physics" split, and remains ready to generate as the direct next course in this Science subject — picking up the same real, verify-everything discipline for electricity, magnetism, and the real crisis that led to relativity.