Acceleration

There is one idea beginners confuse with speed more than any other, and your body already knows the difference. It is not how fast you go. It is how fast the fastness itself is changing.

Sit in a car. Press the gas pedal and you feel pushed back into the seat. A lift starting upward makes you briefly heavier; stopping makes you lighter. A train at a smooth, steady cruise, however fast, feels like nothing at all. What your body reacts to in every one of these is not the speed. It is the change in speed. That is acceleration, and this whole lesson is about pulling it apart from speed for good.

Play first: three graphs, one motion

Same car, now with a third graph stacked underneath: acceleration over time. Switch between a steady cruise, a hard brake, and stop-and-go city driving, then scrub the clock. Watch the acceleration graph carefully. It is flat at zero whenever the speed holds steady, no matter how fast that speed is, and it spikes only when the velocity graph is turning.

Acceleration is the slope of the velocity graph

In the last lesson, velocity was the slope of the position graph. Acceleration is the very same move, applied one level up: it is the slope of the velocity graph. When velocity changes by $\Delta v$ over a time $\Delta t$, the acceleration is their ratio.

Read the profiles through this lens. On a steady cruise the velocity line is flat, its slope is zero, so acceleration is zero, and that stays true whether the cruise is at 30 or 130. During a hard brake the velocity line drops steeply, a large negative slope, so acceleration is a big number pointing backward. That backward-pointing acceleration is the push you feel toward the windshield when a driver stops hard.

There is a mirror fact worth noting: just as the slope of velocity gives acceleration, the area under the velocity graph gives the distance travelled. The two profiles that reach the same top speed but spend different times there cover different ground, and that is the area talking. We use this properly when we simulate motion.

Why your seat pushes you, and your phone knows it

Here is the payoff of "your body feels acceleration, not speed." At a steady 900 km/h in a smooth flight you feel nothing, because nothing is changing. The instant the plane speeds up, slows, or banks, you feel it, because the velocity is changing. A force pressing you into the seat is the felt side of acceleration, which is the first hint of the next lesson's big idea: a force causes acceleration.

Your phone contains a tiny accelerometer that measures exactly this. It reads acceleration, the second change in position, and to recover how fast you are going or where you are it must add those readings up over time, twice. In code, acceleration is the difference of the differences.

vs = [0, 5, 10, 15, 20]      # velocity each second (m/s), flooring it
acc = [vs[i + 1] - vs[i] for i in range(len(vs) - 1)]
# acc == [5, 5, 5, 5]  ->  a steady 5 m/s^2 (velocity climbs 5 every second)

# A phone accelerometer only ever measures 'acc'. Recovering velocity and
# position means summing it back up over time - integration, done twice.

That double sum is why an accelerometer alone slowly loses track of where it is: tiny errors in each reading pile up as you add them, which is exactly why your phone leans on GPS and not just its motion sensors. We meet that drift again in simulating motion.

Check yourself