Foundations — Linear momentum p = mv

1.4.1 · D1 · Physics › Momentum & Collisions › Linear momentum p = mv

Is page par assume kiya gaya hai ki tumhe kuch bhi nahi aata. Parent note Linear momentum $p=mv$ ki ek bhi line trust karne se pehle, tumhe usmein use hone wala har symbol dekhna hoga. Chalte hain ek-ek karke, har ek pichle ke upar rakha hua.

0. Woh picture jis par baar baar wapas aate rahenge

Is topic ki har cheez ek flat sheet par rehti hai — socho ek hockey puck ice par slide kar raha hai, upar se dekha hua. Hum do directions draw karte hain: daayein taraf ko $x$ bolte hain, upar ki taraf ko $y$. Jo bhi quantity milegi, woh is sheet par ek number ya ek arrow hogi.

1. Number vs. Arrow — Sabse Pehla Split

Koi bhi physics se pehle, ek idea decide karta hai ki tum har symbol ko kaise padhoge: kya yeh cheez sirf ek size hai, ya size-with-a-direction?

Topic ko yeh split kyun chahiye: momentum ek vector hai. Agar tum isse plain number maano to collisions galat ho jaayenge, kyunki opposite directions mein move karne wali do cheezein opposite momentum carry karti hain aur cancel ho jaati hain. Neeche poora "signs aur components" machinery sirf isliye hai kyunki momentum ek arrow hai, na ki ek bare number.

2. Chhota Arrow: Notation $\vec{v}$ aur Uski Size $|\vec{v}|$

Zor se padho:

• $\vec{v}$ ::: "velocity vector — kitni tez aur kis taraf"
• $|\vec{v}|$ or $v$ ::: "speed — bas number, koi direction nahi"

Picture: $\vec{v}$ khud arrow hai; $|\vec{v}|$ us arrow ki length hai jo ruler se maapi gayi ho.

Topic ko yeh kyun chahiye: parent note $\vec{p}=m\vec{v}$ hats ke saath likhta hai, aur baad mein $|\vec{p}|=m|\vec{v}|$ unke bina. Yeh alag statements hain — ek arrows ke baare mein, ek lengths ke baare mein — aur tumhe inhe ek nazar mein alag karna aana chahiye.

3. Mass $m$ — "Kitna Stuff Hai" Wala Number

Topic ko yeh kyun chahiye: $m$ momentum ke do ingredients mein se ek hai. Kyunki yeh ek positive scalar hai, vector $\vec{v}$ ko $m$ se multiply karne par arrow stretch hota hai lekin direction kabhi flip nahi hoti — ek fact jis par hum baar baar rely karte hain.

4. Velocity $\vec{v}$ — "Kitni Tez Aur Kis Taraf" Wala Arrow

Topic ko yeh kyun chahiye: velocity momentum ka direction-carrier hai. Kyunki $m$ sirf ek arrow ko stretch kar sakta hai (Section 3), $\vec{p}$ bilkul usi direction mein point karta hai jis mein $\vec{v}$ karta hai. Parent note ka poora "momentum velocity ki direction mein point karta hai" wala claim actually is arrow ke baare mein ek statement hai.

5. Arrow ko Number se Multiply Karna: $m\vec{v}$ Kya Karta Hai

Ab hamare paas $m$ (ek positive number) aur $\vec{v}$ (ek arrow) hain. Formula kehta hai $m\vec{v}$ calculate karo. "Number times arrow" ka matlab kya hota hai?

• Kyunki $m>0$ hamesha hota hai, momentum $\vec{p}=m\vec{v}$ kabhi velocity ke opposite direction mein point nahi kar sakta.
• Result ki length $|\vec{p}|=m|\vec{v}|$ hoti hai — mass times speed.

Topic ko yeh kyun chahiye: yahi actual operation hai jo $\vec{p}=m\vec{v}$ ke andar chhupa hua hai. "Arrow ko scale karo" samajh lene se vector formula mysterious ki jagah obvious ban jaata hai.

6. Arrow ko $x$ aur $y$ Pieces Mein Todna — Components

Sheet par ek single arrow do arrows se rebuild kiya ja sakta hai: ek purely rightward, ek purely upward. Yeh pieces uske components hain.

Topic ko yeh kyun chahiye: kyunki $m$ bas har shadow ko scale karta hai, momentum bhi isi tarah split ho jaata hai: $p_x=m{v}_x,p_y=m{v}_y.$ Yahi tarika hai jisse parent note 2D motion handle karta hai — woh kabhi arrows directly add nahi karta, woh unke $x$-pieces aur $y$-pieces alag-alag add karta hai.

7. Arrow Ki Length Wapas Nikalna: Pythagoras

Jab tumhare paas do shadows $p_x$ aur $p_y$ ho jaayein, original arrow kitna lamba hai?

Topic ko yeh kyun chahiye: parent note ke Worked Example 4 mein $|\vec{p}|=\sqrt{0.6^2+0.8^2}=1.0$ calculate hota hai. Woh ek line yahi triangle hai. Pythagoras ke bina tum components ko magnitude mein wapas nahi badal sakte.

8. 1D Mein Signs $+$ aur $-$ — Direction Ek Line Par Squeezed

Agar sab kuch ek line par move kare, to hume arrows ki zaroorat hi nahi: hum ek direction ko positive choose karte hain aur opposite ko negative.

Topic ko yeh kyun chahiye: Worked Example 3 mein, cart B ki velocity $-2\text{ m/s}$ hai, jis se momentum $-6\text{ kg⋅m/s}$ milta hai; ise $+8$ mein add karne par $+2$ milta hai. Woh signs direction-cancelling kar rahe hain. Inhe miss karo to har collision total galat ho jaayega.

9. Rate of Change — $\frac{d\vec{p}}{dt}$ Padhna

Parent note ka sabse gehara formula $\vec{F}=\frac{d\vec{p}}{dt}$ hai. Ise padhne ke liye tumhe full calculus ki zaroorat nahi.

Topic ko yeh kyun chahiye: Newton's Second Law, impulse, aur Conservation of Linear Momentum ka har zikr is ek rate-of-change symbol ko padhne par depend karta hai. Yeh "kitni jaldi change ho raha hai" ka placeholder hai — kuch zyada scary nahi.

Prerequisite Map

Equipment Checklist

Cover the right side and answer out loud. Agar koi bhi line atka de, parent note kholne se pehle us section ko dobara padho.

Connections

• Linear momentum $p=mv$ — parent topic jisme yeh foundations jaati hain.
• Newton's Second Law — jahan $\frac{d\vec{p}}{dt}$ force ka law ban jaata hai.
• Impulse–Momentum Theorem — rate-of-change symbol ko $\vec{F}dt=d\vec{p}$ mein rearrange karta hai.
• Conservation of Linear Momentum — yahan banaye signs aur components ki zaroorat padti hai.
• Kinetic Energy — momentum ke saath ek aur scalar-vs-vector contrast.