1.4.1
Intuition "Motion ki matra" ka ideaSocho ki ek ping-pong ball aur ek bowling ball — dono same speed se aa rahe hain — tumhe lage. Bowling ball bahut zyada takleef deta hai . Ab wahi bowling ball dheere chal raha ho vs. tezi se phenka gaya ho — tez wala zyada takleef deta hai. Toh kisi cheez ko rokna kitna mushkil hai, yeh do cheezon par depend karta hai milke: uske paas kitna stuff hai (mass) aur woh kitni tezi se ja raha hai (velocity). Momentum bilkul yahi combined "quantity of motion" hai.
Yeh specific combination m v m v 2 m + v
Kyunki Newton ka second law, apne original form mein, p rate of change force ke barabar hoti hai. Isi wajah se yeh "how motion gets passed around" — especially collisions mein — ke liye natural bookkeeping tool hai.
Newton ne apna second law aslaan aise bataya tha: force momentum ki change ki rate hai.
F = d t d p
Aao is ko unpack karein taaki jaana-pehchana F = ma p = m v
Intuition Momentum "conserve" kyun ho sakta haiRearrange karo: F d t = d p hi momentum ki change hai (yeh impulse hai, dekho Impulse–Momentum Theorem ). Agar do objects sirf ek doosre ko push karte hain (internal forces), toh Newton ka third law un pushes ko equal aur opposite banata hai, isliye total p
Kyunki p = m v m
p x = m v x , p y = m v y , ∣ p ∣ = m ∣ v ∣ = p x 2 + p y 2
Intuition Sign matter karta hai1D mein, momentum negative ho sakta hai. 3 m/s ulta hota hai. Jab woh collide karte hain, yeh signs automatically cancelling/adding karte hain tumhare liye — yahi toh point hai ise vector treat karne ka.
Kinetic energy hai K E = 2 1 m v 2
K E = 2 1 m v 2 = 2 m ( m v ) 2 = 2 m p 2
Yeh step kyun? m 2 1 m v 2 = 2 1 m m 2 v 2 = 2 m ( m v ) 2 m v = p
Intuition Do objects momentum share kar sakte hain, energy nahiMomentum linearly v v 2 same momentum ho sakta hai lekin bahut alag kinetic energies. Isliye momentum sabhi collisions mein conserve hota hai lekin KE sirf elastic waalon mein.
Worked example 1) Basic magnitudeEk 0.50 kg 20 m/s p = m v = ( 0.50 ) ( 20 ) = 10 kg⋅m/s, east "East" kyun? Momentum ek vector hai — direction bhi answer ka hissa hai.
Worked example 2) Same momentum, alag massesEk 2 kg 3 m/s 6 kg 1 m/s p 1 = 2 × 3 = 6 kg⋅m/s , p 2 = 6 × 1 = 6 kg⋅m/s K E 1 = 2 1 ( 2 ) ( 3 2 ) = 9 J , K E 2 = 2 1 ( 6 ) ( 1 2 ) = 3 J Yeh kyun matter karta hai: equal momentum ≠ equal energy. Fast light cart 3× zyada energy carry karta hai.
Worked example 3) Negative momentum (1D)Cart A: 2 kg + 4 m/s 3 kg − 2 m/s p A = 2 ( + 4 ) = + 8 , p B = 3 ( − 2 ) = − 6 p total = + 8 + ( − 6 ) = + 2 kg⋅m/s Signs ke saath kyun add kiya? Kyunki yeh ek vector sum hai; opposite directions partially cancel karte hain.
Worked example 4) 2D momentumEk 0.2 kg v x = 3 m/s v y = 4 m/s p x = 0.2 ( 3 ) = 0.6 , p y = 0.2 ( 4 ) = 0.8 ∣ p ∣ = 0. 6 2 + 0. 8 2 = 0.36 + 0.64 = 1.0 kg⋅m/s Pythagoras kyun? Perpendicular components ek right triangle ki tarah add hote hain.
Common mistake "Momentum aur kinetic energy basically same cheez hain."Kyun sahi lagta hai: dono tez jaane par badhte hain, dono rest par zero hain, dono mein m v interchangeable lagte hain .
Fix: p ∝ v K E ∝ v 2 ≥ 0 p K E quadruple ho jaata hai. Dono alag sawaalon ke jawaab dete hain: p K E
Common mistake "Main sirf speeds use karunga aur numbers add kar dunga."Kyun sahi lagta hai: rozmarra ki baat mein hum direction ignore karte hain.
Fix: Momentum ek vector hai. 1D mein tumhe ZAROOR + / −
F = ma F = d p / d t Kyun sahi lagta hai: F = ma Fix: F = d p / d t zyada general hai. Jab mass badlti hai (rockets, badhte huye raindrops, conveyor belts), F = ma F = d p / d t
Recall Feynman: ek 12-saal ke bachche ko samjhaoMomentum "moving cheez mein kitna oomph hai" yeh hai. Kisi cheez mein zyada oomph hota hai agar woh bhaari ho ya tez ho — dono ko multiply karo aur tumhe uska momentum milega. Ek truck aur ek bicycle dono ka same oomph ho sakta hai agar bicycle bahut tez ja rahi ho aur truck rengta ja raha ho. Jab cheezein ek doosre se takraati hain, oomph kabhi gayab nahi hota — woh sirf pass hota rehta hai, jaise marbles share karna. Yahi sharing rule momentum ko itna useful banata hai: crash se pehle total oomph gino, aur baad mein bhi same total paoge.
"Mass Velocity = Punch jo tumhe milega." p = m v
Linear momentum ki definition kya hai? p = m v , ek vector jo velocity ke along point karta hai.
Momentum ki SI units kya hain? kg⋅m/s N⋅s
Kya momentum scalar hai ya vector? Ek vector — iski direction hoti hai (1D mein signs, 2D mein components).
Newton ka 2nd law momentum form mein batao. F = d p / d t F = ma p = m v m F = m d v / d t = ma
F = ma F = d p / d t Jab mass badlti hai (rockets, badhte huye raindrops, conveyor belts).
Kinetic energy ko momentum ke terms mein express karo. K E = 2 m p 2
Do objects ka equal momentum hai lekin alag mass — equal KE? Nahi. Hafla (tez) wale ka zyada KE hoga, kyunki K E = p 2 /2 m
Speed double hone par p K E p v K E v 2
Opposite directions mein chalte objects (1D) ka total momentum kaise nikalte hain? Signs ke saath add karo: opposite directions ko opposite signs milte hain aur yeh partially cancel hote hain.
