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

1.2.2 · D1 · Physics › Newton's Laws & Dynamics › Newton's second law — F = ma (net force), impulse-momentum f

Pehle se ${\vec{F}}_{\text{net}}=\frac{d\vec{p}}{dt}$ padhne ke liye, tumhare paas iska har piece already hona chahiye. Neeche ek chain hai: har rung sirf un rungs se bani hai jo pehle chadh chuke ho. Agar tum last line padh sakte ho, toh parent note padh sakte ho.

1. Sirf size wala number — ek scalar

Picture: ek ruler par ek value. Bas "kitna" — iske aage kuch nahi jaanna.

Yeh topic kyun chahiye: mass $m$ aur time-gap $\Delta t$ scalars hain. Yeh doosri quantities ko scale karte hain upar ya neeche, lekin kabhi kisi taraf point nahi karte.

2. Direction wala number — ek vector

Kuch quantities direction ke bina bekaar hain. "$3$ metre chalo" — kis taraf? "$10$ newtons se push karo" — kahan ki taraf?

Figure dekho. Blue arrow ek vector hai. Uski length (yellow bracket) magnitude hai — kitna bada. Uska tilt direction hai. Wohi arrow kisi aur jagah move karo, same length aur tilt rakho, toh woh same vector hai — sirf size aur direction matter karta hai, kahan baitha hai nahi.

Yeh topic kyun chahiye: velocity $\vec{v}$, acceleration $\vec{a}$, force $\vec{F}$, aur momentum $\vec{p}$ SARE vectors hain. Isliye parent note signs ke baare mein itna obsessive hai — ek ball $+20$ se $-30$ tak slow ho rahi hai toh $10$ nahi badli, $50$ badli, kyunki direction flip hua.

3. Vectors add karna — tip-to-tail

Do pushes ek saath act kar rahe hain. Combined push kya hai? Tum sirf numbers add nahi kar sakte agar woh alag-alag taraf point kar rahe hain.

Figure mein:

• Top row: do forces same direction mein — resultant (green) lamba hai: sizes add hoti hain.
• Middle row: do forces opposite direction mein — partly cancel hote hain; resultant chhota hai aur winner ki taraf point karta hai.
• Bottom row: do forces ek angle par — green resultant unke bane box ka diagonal hai.

Yeh topic kyun chahiye: ${\vec{F}}_{\text{net}}$ vectors ka sum hai. Jab woh ek line par hain (jaise zyaadatar examples mein), hum arrows ko signed numbers se replace karte hain: right $=+$, left $=-$. Tab tip-to-tail addition signed numbers ki ordinary addition ban jaati hai.

4. Rate of change — derivative $\frac{d()}{dt}$

Yeh woh tool hai jis par poora law bana hua hai. Humein ek tarika chahiye yeh kehne ka ki "koi cheez abhi kitni tezi se badal rahi hai."

Maan lo ek quantity $q$ ek chhote time-gap mein badlti hai. Uska average rate of change hai $\frac{\text{change in }q}{\text{change in time}}=\frac{\Delta q}{\Delta t}.$

Figure mein, $q$ ko time ke against plot kiya gaya hai. Red line ek wide gap hai: slope $\Delta q/\Delta t$ us poore stretch ka average hai. Jaise gap shrink karte hain (yellow, phir single green touching line), slope true instant rate par settle hota hai — woh limiting slope derivative hai.

Yeh topic kyun chahiye: Newton's law hai $\vec{F}=\frac{d\vec{p}}{dt}$ — force jis instantaneous rate se momentum badlta hai woh hai. Derivative nahi, law nahi.

5. Velocity, acceleration — derivatives jo tum already jaante ho

Ab hum do sabse important derivatives ko naam de sakte hain.

Picture: velocity woh arrow hai jo dikhata hai tum kahan ja rahe ho aur kitni tezi se; acceleration woh arrow hai jo dikhata hai woh heading-arrow kaise khicha ja raha hai — lamba, chota, ya sideways.

Yeh topic kyun chahiye: parent ka $\vec{F}=m\vec{a}$ woh hai jo $\vec{F}=d\vec{p}/dt$ ban jaata hai jab tum $d\vec{v}/dt$ ko acceleration recognize karte ho.

6. Momentum — "amount of motion" $\vec{p}=m\vec{v}$

Picture: ek heavy truck creeping karta hua aur ek light bullet flying karta hua same momentum carry kar sakte hain — ek lamba-patla arrow aur ek chota-mota arrow equal "punch" rakh sakte hain. Momentum capture karta hai kisi cheez ko rokna kitna mushkil hai.

Yeh topic kyun chahiye: poora law momentum ke baare mein ek statement hai. $\vec{F}=d\vec{p}/dt$.

7. Product rule — $m\vec{v}$ do pieces mein kyun split hota hai

Momentum do cheezein ka product hai jo dono badal sakti hain: $m$ aur $\vec{v}$. Calculus mein ek product ke derivative ke liye ek rule hai.

Picture: ek rectangle socho jiske sides $m$ aur $v$ hain; uska area momentum hai. Ek side badhao toh area ek strip gain karta hai; doosri side badhao toh doosri strip gain karta hai. Do strips do terms hain.

Yeh topic kyun chahiye: yeh exactly woh step hai jo $d\vec{p}/dt$ ko $m\vec{a}+\vec{v}dm/dt$ mein turn karta hai. $dm/dt=0$ set karo (constant mass) aur tumhe $\vec{F}=m\vec{a}$ milta hai. Rakho aur tum rockets handle kar sakte ho — parent ka Example 4.

8. Integral — ek force ko time ke saath add karna

Impulse form derivative ko ulta chalaata hai: "rate of change" ki jagah, hum poochte hain "time ke ek stretch mein total accumulated effect."

Yeh topic kyun chahiye: yeh woh ek line hai jo Impulse–Momentum Theorem produce karta hai, $\vec{J}=\int \vec{F}dt=\Delta \vec{p}$. Ek constant force ke liye area bas ek rectangle hai, isliye integral simple $\vec{F}\Delta t$ ban jaata hai.

Prerequisite map

Equipment checklist

Khud test karo — right side cover karo.

Connections

• Newton's second law — F = ma (net force), impulse-momentum form — woh parent jiske liye yeh page tumhe tayar karta hai.
• Newton's First Law (Inertia) — clearly padhta hai jab "net force = 0" samajh aaye.
• Conservation of Linear Momentum — yahan defined momentum $\vec{p}$ chahiye.
• Variable Mass Systems & Rocket Equation — product-rule split chahiye.
• Work-Energy Theorem — time-integral (impulse) ka space-integral cousin.