Acceleration: When Velocity Itself Changes
The hardest idea in kinematics is that speeding up, slowing down and turning are all the same thing — and that falling is its purest example.
Builds on: 0.1 Position & Velocity: Saying Where, Precisely
The rate of the rate
Velocity says how fast position changes. Acceleration says how fast velocity changes:
The double “per second” trips everyone once. An acceleration of 3 m/s² means: every second, the velocity grows by 3 m/s. After one second, 3 m/s faster; after two, 6 m/s faster. Nothing about it says how fast you are — a jet at cruise has enormous velocity and zero acceleration; a dragster off the line has low velocity and brutal acceleration.
Braking is acceleration too (opposite the motion), and — subtler — so is turning at constant speed, because the direction of the velocity is changing. For now we stay on straight lines; the turning case returns with orbits.
Two formulas carry the unit
Start from rest with constant acceleration, and:
The ½t² is worth staring at: in twice the time you go four times as far, because you are covering the later ground at a higher speed. On a position–time graph, constant acceleration draws a parabola — the curve you met as “curving” last lesson.
Free fall: gravity’s clean experiment
Drop anything near the Earth’s surface and — if air resistance stays small — it gains 9.81 m/s of speed every second. That number is g. Heavy or light does not matter: Galileo’s claim, Apollo 15’s demonstration (a hammer and a falcon feather on the Moon, landing together), and the single most useful constant in this course. You will measure it yourself with string and a stopwatch in Unit 3.
Air resistance is the everyday spoiler: it grows with speed until it balances gravity and the velocity stops climbing — terminal velocity. A skydiver belly-down: about 55 m/s. A feather: centimetres per second, which is why Aristotle got twenty centuries of benefit of the doubt.
You cannot feel velocity — a smooth train at 300 km/h feels like your sofa. You feel acceleration: the push of the seat, the lurch of the lift. That is also why the astronauts on the ISS float: they and their station accelerate toward Earth identically, and what vanishes is the relative push you call weight.
⚗️ Lab — The Drop Tower
Drop a ball, watch v climb and x curve — with or without air.
- In vacuum, check the odometer at 1 s and 2 s: four times the distance in twice the time.
- Turn on air resistance and find the terminal velocity where the speed stops growing.
- Try the Moon’s gravity. Same physics, lazier parabola.
Work it by hand — 0 / 2
No multiple choice here: compute the value and type it. Suffixes like 2.5k or 20m are understood; answers within ±2% count.