Lesson 1: Motion

Everything moves: blood through your veins, trains across China, the Earth around the Sun. Describing motion precisely — how far, how fast, how quickly speed changes — is kinematics, physics’ first chapter. The ancient Chinese tracked celestial motions with exquisite precision (Guo Shoujing’s calendar predicted eclipses to the minute); Galileo rolled balls down ramps to discover acceleration’s laws. This lesson builds the language of motion: distance and displacement, speed and velocity, acceleration, and the equations that predict where moving things will be.

Distance vs. Displacement

Distance is the path length traveled (a scalar — magnitude only); displacement is the straight-line change in position (a vector — magnitude and direction). Walk 3 km east then 4 km north: distance = 7 km, but displacement = 5 km northeast (3-4-5 triangle!). The distinction matters: a marathoner’s displacement is zero (finish = start) despite 42 km of distance.

Position needs a reference frame: "the train is at kilometer 500" means relative to a chosen origin. China’s high-speed rail timetables are displacement schedules — Beijing to Shanghai is 1,318 km of track, and arrival times depend on displacement along the route, not the train’s wandering within stations.

Olympiad-style: an ant walks the edges of a 1 m cube from one corner to the opposite — shortest path? Unfold the cube’s faces: the straight line across two faces has length √5 m. Displacement through space is √3 m (the space diagonal), but the ant must travel on surfaces. Constraints shape paths.

Speed and Velocity

Speed = distance/time (scalar); velocity = displacement/time (vector). A car doing 100 km/h around a circular track has constant speed but changing velocity — direction changes! This is why velocity, not speed, appears in physics’ laws: direction matters.

Average velocity = total displacement / total time; instantaneous velocity = the limit as time intervals shrink (calculus’ derivative!). A speedometer shows instantaneous speed. China’s Fuxing trains cruise at 350 km/h — their velocity vectors point along the track, changing direction through curves.

The classic trap: "A train goes 240 km in 3 hours, then 180 km in 2 hours. Average speed?" Total distance 420 km / 5 h = 84 km/h — NOT (80+90)/2 = 85. Average speed weights by time, not by segment. Always total-over-total.

Acceleration

Acceleration = change in velocity / time — and since velocity has direction, turning at constant speed IS acceleration (centripetal). A car braking from 100 km/h to 0 in 5 s: a = −28 m/s ÷ 5 s... 100 km/h = 27.8 m/s, so a = −5.56 m/s². The negative sign means opposite the motion.

Free fall: Galileo showed all objects accelerate equally under gravity (ignoring air): g ≈ 9.8 m/s² downward. Drop a stone: after 2 s, v = 19.6 m/s; distance fallen = ½gt² = 19.6 m. The ancient Chinese knew falling speeds increase — Mohist texts discuss it — but Galileo quantified it.

High-speed rail acceleration is limited for comfort (~0.5 m/s²): reaching 350 km/h (97 m/s) takes ~200 s and ~10 km! That’s why stations are far apart — physics constrains engineering. Acceleration limits shape the built world.

Equations of Motion

For constant acceleration: v = u + at; s = ut + ½at²; v² = u² + 2as. These three solve every constant-a problem. A ball thrown upward at 20 m/s: max height when v = 0: 0 = 400 − 2(9.8)s ⟹ s ≈ 20.4 m; time to top: t = 20/9.8 ≈ 2.04 s.

Problem-solving ritual: list knowns (u, a, t...), identify the unknown, choose the equation containing it, solve, check units and reasonableness. "A car accelerates from rest at 2 m/s² for 10 s; how far?" s = 0 + ½(2)(100) = 100 m. Systematic, not magical.

Graphs tell the story: position-time slope = velocity; velocity-time slope = acceleration; velocity-time area = displacement. Given a v-t graph, the area under it IS the distance traveled. Graphical methods often beat algebra — learn to read motion graphs fluently.

Projectile Motion

Projectiles combine horizontal uniform motion with vertical accelerated motion — independent! A ball launched at 20 m/s at 30°: v_x = 17.3 m/s (constant), v_y = 10 m/s (decreasing at g). The path is a parabola — Galileo’s discovery, unifying horizontal and vertical.

Range: R = v²sin2θ/g — maximized at 45° (in vacuum). Time of flight: t = 2v sinθ/g. Chinese fireworks — invented with gunpowder (Lesson 19) — are projectile art; modern artillery computers solve these equations thousands of times per second.

Air resistance complicates everything (real projectiles fall short of vacuum predictions), but the independence of horizontal and vertical motion remains the key insight. Sports science — from basketball arcs to javelin throws — is applied projectile physics.

Circular Motion and Relative Velocity

Uniform circular motion: speed constant, velocity turning — centripetal acceleration a = v²/r toward center. A 1,000 kg car rounding a 50 m radius curve at 20 m/s needs a = 8 m/s² — friction provides it (that’s why icy curves are deadly: friction can’t supply the needed v²/r).

Angular quantities: θ (radians), ω = v/r (angular velocity), period T = 2π/ω. A bicycle wheel (r = 0.35 m) at 10 m/s spins at ω ≈ 28.6 rad/s ≈ 273 rpm. Rotational kinematics mirrors linear — same equations, angular variables.

Relative velocity: a boat crosses a river flowing 3 m/s, rowing 4 m/s perpendicular — its ground velocity is 5 m/s at 36.9° downstream (3-4-5 triangle!). Airplanes’ ground speed = airspeed + wind. Reference frames transform velocities by vector addition — Einstein’s relativity began by pushing this idea to light speed.

Deep Dive — Relativity: Motion at Light Speed

Newton’s kinematics works brilliantly — until speeds approach light’s 300,000 km/s. Einstein’s special relativity (1905) revealed: time dilates (moving clocks run slow), lengths contract, and simultaneity is relative. GPS satellites’ clocks tick faster (weaker gravity) yet slower (orbital speed) — engineers correct both relativities, or GPS would drift 10 km per day!

The Michelson-Morley experiment (1887) found light’s speed constant for all observers — no "aether wind." Einstein took this as a postulate: c is invariant. Consequences cascade: E = mc² (mass-energy equivalence), the light-speed limit (nothing with mass reaches c), and spacetime as a unified fabric. Chinese physicists at Beijing’s BESIII and the Jinping lab probe these foundations.

Time dilation is real and measured: muons (cosmic-ray particles, 2.2 μs lifetime) reach Earth’s surface because at near-c speeds, their clocks run slow in our frame — they live ~20× longer. Atomic clocks on airplanes confirm it. The "twin paradox" (traveling twin ages less) is resolved: acceleration breaks the symmetry.

General relativity (1915): gravity is spacetime curvature — mass tells spacetime how to curve, spacetime tells mass how to move. Predictions confirmed: Mercury’s orbit, gravitational lensing (1919 eclipse), gravitational waves (LIGO, 2015 — colliding black holes!). China’s Taiji and TianQin missions will hunt gravitational waves from space.

Relativistic kinematics: momentum p = γmv, energy E = γmc², with γ = 1/√(1−v²/c²). At everyday speeds, γ ≈ 1 — Newton recovered! Relativity contains classical mechanics as a low-speed limit. This is how physics progresses: new theories extend old ones, never merely replacing them.

Why it matters: every high-energy experiment (particle colliders), every GPS fix, every quasar observation uses relativity. And philosophically: space and time aren’t absolute backdrops but participants in physics. Motion’s deepest description — from Galileo’s ramps to Einstein’s spacetime — is humanity’s greatest intellectual journey.

Conclusion — Why This Matters

Motion is the most visible face of physics — from a sprinter’s dash to a satellite’s orbit, everything that moves obeys the kinematic relationships in this lesson. The Chinese high-speed rail network, with trains cruising at 350 km/h, is applied kinematics: precise control of velocity and acceleration over thousands of kilometers. Master describing motion, and dynamics (Lesson 2) will explain its causes.

Going deeper — reading a journey from its velocity–time graph

Graphs compress a whole journey into one picture, and examiners love them because every feature means something physical. Study this worked example. A delivery e-bike in Shenzhen accelerates uniformly from rest to 20 m/s in 5 s, cruises at 20 m/s for 10 s, then brakes uniformly to a stop in 4 s.

• Acceleration (first phase): the slope of the v–t graph, a = Δv/Δt = 20/5 = 4.0 m/s². The braking phase has slope −20/4 = −5.0 m/s² — negative because velocity decreases.
• Displacement: the area under the graph. Triangle (accelerating): ½ × 5 × 20 = 50 m. Rectangle (cruising): 10 × 20 = 200 m. Triangle (braking): ½ × 4 × 20 = 40 m. Total = 290 m.
• Average speed: 290 m ÷ 19 s ≈ 15.3 m/s — note it is NOT the average of 0 and 20 (that shortcut only works for a single phase of uniform acceleration).

The same area rule works for any shape: count squares under a curved v–t graph and you have the displacement, no equations needed. This is why Lesson 1 calls graphs "the story of the motion": slope tells you acceleration, area tells you displacement, and the sign of the slope tells you whether the object is speeding up or slowing down.

Practical focus — measuring g with a falling ball and light gates

Method. A steel ball is released from rest and falls 0.500 m through two light gates, which time the fall. Three trials give 0.322 s, 0.318 s, 0.320 s.

Analysis. Mean time t = 0.320 s. From s = ½gt², g = 2s/t² = 2 × 0.500 / 0.320² = 1.000/0.1024 ≈ 9.77 m/s². Spread of trials (±0.002 s, about 0.6%) propagates to about ±1.2% in g, so g = 9.77 ± 0.12 m/s² — the accepted 9.80 lies inside the range.

Evaluation. The value sits slightly low, as expected: air resistance and any release mechanism delay (the clock starting before the ball truly moves) both lengthen t. Improvements: repeat and average; use a taller drop (same delays become a smaller fraction); release electromagnetically with the timer triggered by the release itself.

Common mistakes

• Averaging speeds the wrong way. Average speed is total distance ÷ total time — never the mean of two speeds unless the times are equal.
• Swapping distance and displacement on any path that turns or returns.
• Dropping signs: deceleration is acceleration opposite to the velocity; the minus sign is information, not decoration.

Exam Experience

Gaokao-style practice — work under time pressure, then mark yourself strictly with the scheme.

Question 1 (choice, 3 marks). An athlete runs exactly one lap of a 400 m track, finishing where she started. Her distance and displacement are:

• A. 400 m, 400 m
• B. 400 m, 0 m
• C. 0 m, 400 m
• D. 0 m, 0 m

Answer: B. Distance is the full path length (400 m); displacement is final minus initial position — she ends at the start, so it is zero. A confuses the two definitions.

Question 2 (choice, 3 marks). A v–t graph is a straight line from (0, 0) to (10 s, 30 m/s). The displacement in the 10 s is:

• A. 75 m
• B. 150 m
• C. 300 m
• D. 600 m

Answer: B. Displacement = area under the graph = ½ × 10 × 30 = 150 m. C is the trap: it multiplies 10 × 30 as if the speed were constant.

Question 3 (structured, 6 marks). A Fuxing train accelerates uniformly from rest to 300 km/h with acceleration 0.55 m/s². (1) Find the time taken. (2) Find the distance covered while accelerating. (3) State the average speed during this phase, and one reason real trains limit acceleration.

Marking scheme. (1) 300 km/h = 83.3 m/s [1]; t = v/a = 83.3/0.55 ≈ 152 s [1]. (2) s = ½at² = ½ × 0.55 × 152² ≈ 6.3 × 10³ m ≈ 6.3 km [1 formula, 1 value]. (3) Average speed = 83.3/2 ≈ 41.7 m/s (≈150 km/h) for uniform acceleration from rest [1]; reason: passenger comfort / wheel–rail adhesion / power limits [1]. No mark for a correct value with no working — Gaokao marking pays for steps.