Physics I: Mechanics — Three Ways to Keep the Books

A car doing 50 km/h stops in about 14 metres on a dry road. Most people guess that at 100 km/h it takes about 28. It takes about 56. The extra speed doesn't just cost twice the distance, it costs four times, and the same fact is why a crash at 100 km/h is not "twice as bad" as one at 50.

Nobody needs a formula to be afraid of that, but a person who has done a first mechanics course sees it: the brakes have to get rid of the car's energy, energy grows with the square of speed, and the road can only take it away at a steady rate per metre. That one chain of thought is most of the course.

Physics I is less a list of topics than a way of keeping accounts. There are three ledgers, force, energy and momentum, and a good physicist picks the one in which the problem is easiest. If you want an easier first read, the library has The Four Forces (13/01), Energy: The One Currency (13/02) and Orbits (13/11). This piece goes further: it makes you use the ledgers.

Newton's laws are one accounting system

Isaac Newton set out his three laws in the Principia in 1687. Read as a bookkeeper would, they say: momentum (mass times velocity) is the thing being tracked; a force is a deposit or withdrawal; and every deposit has a matching withdrawal somewhere else.

The first law says that with no net force the account doesn't change: things keep their velocity. The second says that a net force changes momentum at a steady rate, which for a fixed mass is the famous rule below. The third says forces come in pairs between two bodies: if you push a wall, the wall pushes you, and the pair never acts on the same body, which is why they don't cancel.

Worked number: a 1,500 kg car getting from 0 to 100 km/h (27.8 m/s) in 8 seconds accelerates at 3.47 m/s², so the road must push it forward with about 5.2 kN. Two ice skaters of 50 kg and 70 kg who push apart from rest leave with equal and opposite momentum: if the heavier moves at 1.0 m/s the lighter goes at 1.4 m/s the other way. Total momentum stays at zero.

Try the classic: you on a scale in an elevator. The scale doesn't read your weight, it reads the force pushing up on you.

Accelerating upward at 2 m/s² the scale reads about 827 N, like 84 kg. At constant speed it reads your true weight, whether the elevator is moving or still. In free fall it reads zero. That is exactly the "weightless" astronaut of Orbits.

Free-body diagrams, and friction as a force that only pushes as hard as it must

The graduate's first reflex on any problem is to draw one object, alone, and every push on it: weight, the surface pushing back, friction, tension. Nothing else. That is a free-body diagram. Add the forces as vectors, split them along the directions that matter, and F = ma does the rest.

Friction is the force beginners get wrong. Static friction is not a fixed number: it pushes exactly as hard as needed to stop sliding, up to a ceiling of μ_s times the normal force (how hard the surface pushes back). Only once the ceiling is passed does the block slide, and then friction drops to μ_k × normal force.

On a ramp the weight splits into a part along the slope, mg sin θ, and a part pressing into it, mg cos θ. Sliding begins when the first beats μ_s times the second, which gives tan θ = μ_s. The mass cancels. Check that in the tool: change the mass and nothing about when it slips changes. (The coefficients are round textbook values; real surfaces vary a lot.)

With μ_s = 0.5 the block holds until 26.6° and then slides. At 40° it accelerates down at about 4.05 m/s² (with μ_k = 0.3). This is why a pile of sand has a steepest slope, and why designers of ramps, screws and wedges care so much about one number.

Energy: the balance that doesn't care about the path

Force problems get painful when the path is complicated. Energy is the escape. The work a force does is force times distance moved along it, and the work–energy theorem says the net work on a body equals its change in kinetic energy. When only gravity or springs are involved, the total of kinetic plus potential energy is simply conserved, whatever the route.

A bobsled dropping 30 m ends at √(2 × 9.81 × 30) = 24.3 m/s (87 km/h) with no friction, however many bends the track has. Now back to the opening: the brakes do work μ × m × g × d against the car's ½mv², so the stopping distance is v² ÷ (2μg). Double v, quadruple d.

The sharpest result: brake from 100 km/h with an obstacle 14 m away, the distance that stops a 50 km/h car on a dry road. You still hit it at about 87 km/h, with 75% of the energy you started with: 14 metres of hard braking has removed only a quarter of it.

Momentum: what survives every crash

Energy has a weakness: in a collision much of it turns to heat, sound and bent metal, so the energy ledger is no use. Momentum, though, is conserved in any collision, because the two forces are equal and opposite for the same short time (the third law again). The total before equals the total after, with direction as a sign.

How bouncy the collision is gets one number, the restitution e: 1 for perfectly bouncy, 0 for stuck together. Momentum doesn't care about e. Kinetic energy does.

With equal masses and e = 1, the moving cart stops dead and the other leaves with its speed: the toy known as Newton's cradle. Stuck together, two 1,000 kg cars (one at 10 m/s, one parked) leave at 5 m/s and half the kinetic energy, 25 kJ, becomes damage. That lost half is the reason cars have crumple zones: they spend the energy deforming metal instead of people.

Rotation: the same laws with new names

Everything above has a spinning twin. Force becomes torque, force times the perpendicular distance from the pivot: 80 N on a 0.25 m wrench is 20 N·m, and a longer wrench gives more for the same push. Mass becomes moment of inertia I, which counts not just how much mass there is but how far from the axis it sits. Momentum becomes angular momentum L = I × ω.

Mass far from the axis counts for much more, because I grows with the square of distance. That's why a solid disc beats a hoop of the same mass and radius down a ramp: the hoop keeps all its mass at the rim, so it spends more of its energy on spinning (the disc's acceleration is two thirds of g sin θ, the hoop's one half, whatever the size or mass). And it's why a skater spins faster when she pulls her arms in.

From arms out to tucked in, I falls from 3.15 to 0.99 kg·m², so the spin rises from 1 to about 3.2 turns per second. The energy also rises, from 62 J to 198 J, and the difference is the work her muscles do pulling the weights inward.

Circular motion and gravity: a force that points at the middle

Going round a bend at constant speed is still accelerating, because the velocity keeps changing direction. The acceleration points to the centre and equals v²/r, so something must push sideways with force mv²/r. On a flat road that something is friction. No more friction, no more turn: the car continues in a straight line, which feels, from inside, like being thrown outward.

Notice the engine's power isn't in the formula. On a 50 m bend with μ = 0.5 (wet) the limit is 56 km/h; dry (0.7), 67 km/h; ice (0.1), 25 km/h.

Replace friction with gravity and you get orbits. Set v²/r equal to GM/r² and the period T satisfies T² ∝ r³, Kepler's third law. Check it: the Space Station's lap of 92.6 minutes at 6,779 km from Earth's centre predicts, for the Moon at 384,400 km, a period of 27.5 days. The real month is 27.3. One law, from a bend in a road to the Moon.

Which ledger? Choosing force, energy or momentum

This is the real skill. A hard problem in one ledger is a one-liner in another. Three rough rules: if the question asks for a force or a time, use F = ma; if it asks for a speed between two positions, and the path is messy, use energy; if there's a collision or explosion, use momentum. If something spins, use angular momentum. Try five.

What a graduate just sees

• Stopping distance goes with speed squared: double the speed, four times the distance.
• A scale reads the push on you, not your mass: it changes in an accelerating elevator.
• Static friction is only as large as it has to be, up to a limit that depends on μ and the pressing force.
• Energy ignores the path; use it when the route is complicated.
• Momentum survives every collision; kinetic energy rarely does.
• Mass far from the axis counts for much more in rotation, because distance is squared.
• A turn needs a sideways force; remove the grip and you go straight.
• Weightless means falling together, not "no gravity".

Where it shows up next

• Statics (34/10) is F = ma with a = 0: the free-body diagram becomes the bridge or crane that stays put.
• Dynamics (34/12) and Vibrations (34/17) take rotation and the second law into machines and springs, usually as differential equations (Differential Equations, 34/03).
• Thermodynamics (34/14) is the energy ledger with heat added: the lost half of that crash is where it starts.
• Physics II (34/07) redoes the same accounting for charges, with Coulomb's force and voltage as energy per charge.
• In Programming Fundamentals (34/09), you'll turn F = ma into a simulation loop; in Computer Science game engines and robotics code, it's the physics step.

If you want to go deeper

• 📘 Standard textbooks. Halliday, Resnick & Walker, Fundamentals of Physics (12th ed., Wiley, 2021), or Young & Freedman, University Physics (16th ed., Pearson); the usual first-year choices, with worked examples and thousands of problems. For a sharper, more mathematical route, Kleppner & Kolenkow, An Introduction to Mechanics (2nd ed., Cambridge).
• 📘 Gentler, free. The Feynman Lectures on Physics, Volume I, free online at feynmanlectures.caltech.edu; chapters 4 (energy) and 10 (momentum) are a good start.
• 🌐 Free course. MIT OpenCourseWare 8.01SC Classical Mechanics (Fall 2016): lectures, problem sets and solutions.