Force, mass, and acceleration

An empty shopping cart springs forward at the lightest touch. Load it with a week of groceries and the same shove barely stirs it. A loaded truck needs far more road to stop than an empty one. There is a single equation under all of it.

The last lesson said a net force changes an object's motion but not by how much. This lesson fills in the "how much," and it turns out to be one of the most useful sentences in all of physics. Two things decide how sharply the motion changes: how hard you push, and how much stuff you are pushing.

Play first: push, mass, and the acceleration between them

Set an applied force and a mass, and read the acceleration off the panel before you even press Push. Now do the key experiment: hold the force fixed and drag the mass up. Watch the acceleration number fall. Then turn on friction and notice that only the leftover, the net force, drives the box.

Two patterns fall out of playing with it. Push twice as hard, at fixed mass, and the acceleration doubles. Double the mass, at fixed push, and the acceleration halves. Acceleration grows with force and shrinks with mass.

F = ma, built from those two patterns

Write those two observations as proportionalities. Acceleration is proportional to the net force, and inversely proportional to the mass.

Combine them into one relation and choose units (the newton) so the constant of proportionality is exactly one, and you get Newton's second law, the exchange rate between push, heft, and change of motion.

Mass is inertia, not weight

Notice what mass is doing in that equation. It sits in the denominator of $a=F/m$, dividing the force down. A bigger mass resists the same force more. That is exactly the inertia from the last lesson, now given a number. Mass is the measure of inertia.

This is not the same as weight, even though we weigh things in daily life to find their heft. Mass is how much an object resists being accelerated; it is the same on Earth, on the Moon, or drifting in deep space. Weight is the pull of gravity on that mass, and it changes with where you are. We keep them apart carefully when we reach gravity. For now: the $m$ in $F=ma$ is inertia, not weight.

The same law is the heart of a physics engine

Every game and simulation that moves objects runs this exact rule, once per frame. Add up the forces on a body, divide by its mass to get acceleration, then use that acceleration to update velocity and position. That last step, turning acceleration into motion over time, is integration, and it is the whole subject of a later lesson.

mass = 2.0                        # kg
force = 6.0                       # net newtons, right now
a = force / mass                  # a = F / m  ->  3.0 m/s^2

# advance the motion one small timestep (this stepping is 'integration')
dt = 0.016                        # ~one frame at 60 fps
velocity += a * dt                # acceleration changes velocity
position += velocity * dt         # velocity changes position

That is the engine of simulating motion, where the small print of how you take that step turns out to matter enormously. And because the net force is what counts, the moment two objects push on each other, we need interaction pairs and friction to see where those forces come from.

Check yourself