Impulse-momentum theorem — derivation — Physics notes

WHAT are we even talking about?

WHY do we care? Because in collisions, explosions, kicks, and catches the force is huge, messy, and lasts a tiny time. We almost never know $\vec{F}(t)$ in detail — but we can measure how velocity changed. The theorem lets us connect the two.

HOW to derive it — from Newton's 2nd law (first principles)

Newton's second law in its original form (the form Newton actually wrote) is:

$\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}$

Why this form, not $F=ma$ ? Because $\vec F = m\vec a$ secretly assumes mass is constant. The momentum form is more fundamental.

Step 1 — Start from the rate of change of momentum. $\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}$ Why this step? This is the definition of force as momentum-changer; everything follows from here.

Step 2 — Multiply both sides by $dt$ (separate the differentials). $\vec{F}_{\text{net}}\,dt = d\vec{p}$ Why this step? We want total effect over a time interval, so we prepare to add up tiny pushes.

Step 3 — Integrate over the interval $t_1 \to t_2$ . $\int_{t_1}^{t_2}\vec{F}_{\text{net}}\,dt = \int_{\vec{p}_1}^{\vec{p}_2} d\vec{p}$ Why this step? The left side adds up every tiny push; the right side just adds up tiny changes in momentum.

Step 4 — Evaluate the right-hand integral. $\boxed{\vec{J} = \int_{t_1}^{t_2}\vec{F}_{\text{net}}\,dt = \vec{p}_2 - \vec{p}_1 = \Delta\vec{p}}$ Why this step? The integral of $d\vec p$ is just the final minus initial momentum.

Figure — Impulse-momentum theorem — derivation

Worked Examples

Worked example 1 — Catching a ball

A $0.15\,\text{kg}$ ball hits your glove at $20\,\text{m/s}$ and stops in $0.05\,\text{s}$ . Find the average force.

Step 1: $\Delta p = m v_f - m v_i = 0.15(0) - 0.15(20) = -3\,\text{kg·m/s}$ . Why? Final velocity is 0 (caught); the sign shows force opposes the ball's motion.

Step 2: $\bar F = \dfrac{\Delta p}{\Delta t} = \dfrac{-3}{0.05} = -60\,\text{N}$ . Why? Theorem: impulse = $\bar F\,\Delta t = \Delta p$ , so $\bar F = \Delta p/\Delta t$ .

Step 3 (the insight): If you "give" with your hand, $\Delta t$ goes up → $\bar F$ goes down → it hurts less. Why? $\Delta p$ is fixed (ball must stop), so stretching time shrinks force.

Common Mistakes (Steel-manned)

Recall Feynman: explain to a 12-year-old

Imagine pushing a swing. If you push hard for a split second, the swing speeds up a little. If you push gently but keep pushing for a long time, it can speed up just as much! The total "push spread over time" is called impulse, and it exactly equals how much the swing's "moving-ness" (momentum) changes. That's also why catching an egg gently — moving your hands back as you catch — keeps it from breaking: same change in motion, but spread over more time means a gentler force.

Active Recall

What is the impulse–momentum theorem in words?

The net impulse on an object equals its change in momentum,

\vec J = \Delta\vec p

From which law is the theorem derived?

Newton's 2nd law in momentum form,

\vec F = d\vec p/dt

, integrated over time.

What is the formula for impulse with a time-varying force?

\vec J = \int_{t_1}^{t_2}\vec F\,dt

— the area under the force–time graph.

Units of impulse, and why they match momentum?

\text{N·s} = \text{kg·m/s}

; since

J = \Delta p

, they must be the same unit.

How do you find average force from impulse?

\bar{\vec F} = \Delta\vec p / \Delta t

Why does a bouncing ball deliver more impulse than one that stops?

Stopping gives

\Delta p=-mv

; bouncing gives

\Delta p=-2mv

(velocity reverses), doubling the impulse.

Why do airbags reduce injury?

They increase

\Delta t

, so for the same

\Delta p

the average force

\bar F=\Delta p/\Delta t

drops.

For constant force, impulse simplifies to?

\vec J = \vec F\,\Delta t

Connections

Newton's Second Law — the parent law this is derived from.
Conservation of Momentum — what happens when net external impulse is zero.
Collisions — Elastic and Inelastic — uses impulse to analyze contact forces.
Force–Time Graphs — impulse = area under the curve.
Work–Energy Theorem — the spatial analogue (force over distance) vs impulse (force over time).
Center of Mass Motion — external impulse moves the system's total momentum.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, idea bahut simple hai: kisi cheez ki "moving-ness" ko hum momentum kehte hain, $p = mv$ . Ab agar tum us cheez par force lagao kuch time tak, toh uska momentum badal jaata hai. Force ko time ke saath jod do (integrate karo) — usko impulse kehte hain, $J = \int F\,dt$ . Aur theorem kehta hai: impulse exactly equal hota hai momentum ke change ke, yaani $J = \Delta p$ . Bas itni si baat — par exam mein 80% impulse/collision problems isi ek line se ban jaate hain.

Ye nikalta kahan se hai? Newton ka second law ki asli form hai $F = \frac{dp}{dt}$ . Dono taraf $dt$ se multiply karo, fir $t_1$ se $t_2$ tak integrate kar do — left side ban jaata hai impulse, right side ban jaata hai $\Delta p$ . Done. Koi formula ratne ki zaroorat nahi, derivation khud yaad rakho.

Real life connection: cricket ball catch karte time hum haath peeche kheechte hain. Kyun? Kyunki ball ko rokne ke liye $\Delta p$ toh fixed hai, par agar $\Delta t$ bada kar do toh $\bar F = \Delta p / \Delta t$ chhota ho jaata hai — chot kam lagti hai. Yahi airbag aur car ke crumple zone ka funda hai.

Ek important trap: agar ball bounce karti hai toh impulse double ho jaata hai compared to sirf rukne ke. Kyunki velocity ka direction ulta ho jaata hai, $\Delta p = -2mv$ banta hai. Isiliye signs ka dhyaan rakho — impulse ek vector hai, sirf magnitude mat lagao.

Impulse-momentum theorem — derivation

WHAT are we even talking about?

HOW to derive it — from Newton's 2nd law (first principles)

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Momentum & Collisions

Connections