Newton's second law — F = ma (net force), impulse-momentum form: Visual walkthrough

We assume you know only three plain ideas: an object has a speed and a direction (that arrow is called velocity), things have a heaviness (mass), and a force is a push or pull. Everything else — momentum, rate of change, area under a graph, impulse — we build here.

Step 1 — What is momentum? (the "amount of motion")

WHAT we did: multiplied a number ( $m$ ) by an arrow ( $v$ ) to get a longer arrow ( $p$ ).

WHY we bother: "how hard is this thing to stop?" depends on both speed and heaviness. A slow truck and a fast pebble can be equally troublesome. Momentum is the single quantity that bundles both.

PICTURE: in the figure, the thin blue arrow is velocity $v$ ; the thick pink arrow $p$ is that same arrow scaled up by the mass. Same direction, bigger length.

Figure — Newton's second law — F = ma (net force), impulse-momentum form

Step 2 — Force is what changes momentum

We are handed one law of nature (this is the true form of the second law):

WHY the tool $\frac{d}{d t}$ ? We need to talk about change, and not just total change but change per second. That "per second, right now" idea is exactly what the derivative $\frac{d}{d t}$ was invented to measure. No other tool answers "how fast is this arrow shifting at this instant?"

WHAT it says: point a force at something, and its momentum arrow starts sliding in the direction of that force. No force ⇒ the arrow is frozen ⇒ velocity constant (that's Newton's First Law (Inertia) hiding inside).

PICTURE: the momentum arrow at time $t$ is chalk-blue; a moment later ( $t + d t$ ) it has grown a small pink tip $d p$ pointing along the force $F$ .

Step 3 — Rearrange: a tiny push gives a tiny momentum change

Multiply both sides of $F = \frac{d p}{d t}$ by the small time slice $d t$ :

$F_{net} d t = d p$

WHAT each symbol is now doing:

$F_{net} d t$ — force held over a sliver of time $d t$ . A little push-for-an-instant.
$d p$ — the little chunk of momentum that little push delivered.

WHY we did it: we want to accumulate the effect of a force over a whole stretch of time. The trick is: chop time into many tiny slices $d t$ ; each slice hands the object one small momentum chunk $d p$ . Add them all up. That "add up infinitely many tiny pieces" is precisely what an integral does — so an integral is the tool we reach for next.

PICTURE: a single thin vertical strip on a Force-vs-time graph. Its width is $d t$ , its height is $F$ , and its area $F d t$ equals the momentum chunk $d p$ .

Step 4 — Add up every slice: the integral

Sum (integrate) every sliver from a start time $t_{1}$ to an end time $t_{2}$ :

$\int_{t_{1}}^{t_{2}} F_{net} d t = \int_{t_{1}}^{t_{2}} d p$

The right-hand side is the easy one: adding up all the tiny changes in $p$ just gives the total change, start value subtracted from end value.

$\int_{t_{1}}^{t_{2}} d p = p_{2} - p_{1} = Δ p$

WHAT the pieces mean:

$\int_{t_{1}}^{t_{2}} F_{net} d t$ — total area under the force–time curve (the sum of all the thin strips).
$p_{2}$ — momentum at the end. $p_{1}$ — momentum at the start.
$Δ p$ — read "delta pee", meaning "the change in $p$ ": final minus initial.

WHY the right side collapses so neatly: this is the Fundamental Theorem of Calculus in disguise — chopping a change into tiny pieces and re-adding them recovers the whole change. The middle terms all cancel.

PICTURE: many strips filling the whole coloured area under the curve on the left; on the right, two momentum arrows with the small difference arrow $Δ p$ between their tips.

Step 5 — Name the left side: Impulse

WHAT we've proven: the whole shaded area of a force–time graph is the change in momentum. Two very different-looking things — an area and an arrow-difference — are the same number.

WHY this is a big deal: it lets us skip the messy details of how the force wobbled during a hit. Only the area matters. A gentle 5 N for 2 s and a violent 1000 N for 0.01 s can hand over the identical momentum change if their areas match.

PICTURE: an ugly, spiky real-world force pulse (like a bat hitting a ball) and a neat rectangle of the same area beside it. Same impulse, same $Δ p$ .

Step 6 — The constant-force shortcut (the easy special case)

If the force does not change during the interval, the area under the graph is just a rectangle: height $F$ times width $Δ t$ .

$J = F Δ t = Δ p$

$F$ — the (steady) force. $Δ t = t_{2} - t_{1}$ — how long it acts.
Their product is the rectangle's area.

WHY keep this case separate: most textbook problems (a constant kick, an average force) live here. It turns a scary integral into ordinary multiplication.

PICTURE: a flat-topped rectangle — the simplest possible strip-sum.

Step 7 — Trade-off: same area, wide vs tall (why airbags work)

The theorem says only the area $J$ is fixed by the crash ( $Δ p$ is set by your mass and speed). So you get to choose the shape of the rectangle:

$F = \frac{Δ p}{Δ t} \Rightarrow bigger Δ t \Rightarrow smaller F .$

WHAT: widen the time base $Δ t$ and the same area forces the height $F$ down.
WHY it saves lives: injury is about force (height), not impulse (area). Airbags, crumple zones, and bending your knees all stretch $Δ t$ .

PICTURE: two rectangles of equal area — one tall-and-thin (hard dashboard, huge $F$ ), one short-and-wide (airbag, small $F$ ).

Step 8 — The degenerate cases (never get caught out)

PICTURE: a force curve that dips below the axis — the pale region above adds, the blue region below subtracts; the net impulse is the difference.

The one-picture summary

Everything on this page is one chain: force over time = area = change in momentum arrow. The figure below stacks all three views — the tiny strip, the full area, and the two momentum arrows with their difference — into a single diagram.

Recall Feynman retelling — the whole walkthrough in plain words

Momentum is just "amount of motion" — heaviness times how-fast-and-which-way. A force is the thing that changes that amount, and how much it changes each second is the derivative $\frac{d p}{d t}$ . If you multiply a little push by a little bit of time, you get a little chunk of momentum handed over — that chunk is the area of one thin strip on a force-versus-time graph. Add up every strip across a whole event and you've got the total area, which equals the total change in the motion arrow. We call that area "impulse." So impulse is the momentum change — same thing, drawn two ways. If the push is steady, the area is a plain rectangle, height-times-width. And the clever bit: the area is fixed by the crash, but you get to pick the rectangle's shape — spread it wide over lots of time and the height (the dangerous force) shrinks. That's every airbag, every knee-bend, every caught cricket ball. Special cases: no push means no area means motion never changes; a push that flips direction has its area partly cancel; and if the mass itself is changing, you must track the momentum arrows directly, because $F = ma$ quietly lies.

Connections

Parent: Newton's 2nd Law — the result this page derives visually.
Newton's First Law (Inertia) — the zero-area case.
Conservation of Linear Momentum — what zero net impulse guarantees.
Collisions and Elasticity — impulse applied during contact.
Friction — a common signed contributor to the area.
Variable Mass Systems & Rocket Equation — where you must use $Δ p$ , not $m Δ v$ .
Work-Energy Theorem — the space-integral of force (impulse is the time-integral).