0.1 Pizza Delivery Driver in a Rush

An introductory scenario for the book’s playground: a delivery vehicle as a point mass, pushed by a force the driver control with the throttle pedal.

The scene

A driver carries pizzas from the shop to a customer’s door. Distance $D$ along the street(s) is known. Income favors short delivery time — but the pizzas (and the brakes) care how the trip is driven.

We strip the world down. No hills, no lane changes, no fuel cost. Position along the road is a single number $x$. Speed is $\dot{x}$ (also written $v$). The driver applies a force $F$ through the throttle. For this first page that force is open-loop: pick $F$ and hold it. There is no feedback yet, and no braking model — on purpose. The point is to feel Newton’s law before we close the loop in later modules.

Newton’s second law

Isaac Newton’s second law says that the net force on a body equals mass times acceleration:

or, solving for acceleration,

Mass $m$ (kilograms) measures how stubborn the vehicle is about changing its motion. A loaded bike has larger $m$ than a light scooter.

Force $F$ (newtons) is what the drivetrain pushes with. More throttle means larger $F$ (in this idealized model).

Acceleration $a$ (metres per second squared) is how quickly speed changes. Same shove on a heavier bike → smaller $a$. Same bike with a harder shove → larger $a$.

For our 1-D delivery run with no drag,

Acceleration and speed

Acceleration is not speed. Speed $v$ says how fast you are moving right now. Acceleration says how fast that speed is changing.

• If $a > 0$ and you keep the force on, $v$ climbs.
• If $a = 0$ (coast with no drag in this model), $v$ stays constant.
• Distance traveled is the accumulation of speed over time: you get to the door by integrating $v$, not by staring at $a$ alone.

In symbols, starting from rest ($v(0) = 0$, $x(0) = 0$) with constant $a$:

So a larger $a = F/m$ means you cover distance $D$ sooner — but you also arrive with higher speed $v = \sqrt{2 a D}$. Without a plan to brake, “faster delivery” and “gentle arrival” fight each other. That tension is why control (not just raw force) will matter later.

Back to the pizza driver

Try a light scooter versus a loaded bike with the same force: the loaded run accelerates less, takes longer, and hits the door slower. Try a hard shove on a short block: you get there quickly with a lot of leftover speed — great for the clock, rough on the pizza.

The widget below is that open-loop experiment. Pick a scenario or drag the sliders, then Run. Physics steps at a fixed $\Delta t$; the drawing only samples the state for display.