Perfectly inelastic collisions — maximum KE loss: Foundations

1.4.7 · D1 · Physics › Momentum & Collisions › Perfectly inelastic collisions — maximum KE loss

Parent note Perfectly inelastic collisions ko padhne se pehle, har ek symbol ka matlab tumhe pata hona chahiye. Neeche, hum har symbol ko scratch se build karte hain — pehle plain words mein, phir ek picture, phir kyun is topic ko iski zaroorat hai. Upar se neeche padho; har brick pichli wali ke upar tiki hai.

1. Mass — $m$

Picture. Ek shopping trolley socho. Ek khaali trolley (chhota $m$ ) ko hilana ya rokna aasaan hai. Eenton se bhari trolley (bada $m$ ) dono ka resistance karti hai. Same push, alag response — woh resistance hi mass hai.

Topic ko iski zaroorat kyun hai. Poora chapter do bodies ke baare mein hai, isliye hum unhe $m_{1}$ aur $m_{2}$ lete hain. Subscripts $1$ aur $2$ sirf name tags hain — "object one" aur "object two" — kuch mathematical nahi. Jab wo chipakti hain, hum unhe add karte hain: $m_{1} + m_{2}$ combined blob ki mass hai.

2. Velocity — $u$ , $v$ , aur crucial minus sign

Do letters kyun? Pure convenience: $u$ = "pehle" (alphabet mein u, v se pehle aata hai), $v$ = "baad mein". Toh $u_{1}$ object 1 ki velocity hai impact se pehle; $v$ wo shared velocity hai jab wo chipak jaate hain.

Picture — direction ek sign hai. Ek seedhi line par, ek direction ko $+$ maano (maano, rightward). Left ki taraf jaane wali koi bhi cheez minus sign leti hai. $- 6$ m/s par ek ball left ki taraf $6$ m/s se chal rahi hai.

Figure — Perfectly inelastic collisions — maximum KE loss

3. Momentum — $p = m v$

Multiply kyun karein, add kyun nahi? Ek truck walking pace par aur ek bullet dono ko rokna mushkil hai, lekin opposite reasons se — huge $m$ vs huge $v$ . Multiply karna dono ko ek number mein blend kar deta hai. Ek bada- $m$ slow object aur ek chhota- $m$ fast object ka same momentum ho sakta hai.

Picture. Har object ke momentum ko ek arrow se represent karo: iski length $∣ m v ∣$ hai (kitna bada) aur iski direction hai wo kis taraf point karta hai. Do objects ko combine karne ke liye, unke arrows ko tip-to-tail rakh do aur total padho.

4. Kinetic energy — $K E = \frac{1}{2} m v^{2}$

Square kyun, aur $\frac{1}{2}$ kyun? Square Work–Energy theorem se aata hai: kisi moving object ko rokne ke liye tumhe work remove karni padti hai, aur "force over stopping distance" ka maths $v^{2}$ deta hai. Speed double karo toh KE chaar guna ho jaati hai. $\frac{1}{2}$ wo constant hai jo wo theorem humein deta hai — abhi ke liye ise accept karo.

Picture — do axes do alag stories sunate hain.

5. Change aur loss — $Δ$ aur $Δ K E$

Picture. Agar tumhare paas pehle $27$ J motion-energy thi aur baad mein $0$ J, toh $Δ K E = 27$ J universe se gaayab nahi hui — wo heat, sound, aur clay ki permanent squish mein badal gayi. $Δ$ sirf wo subtraction hai jo us leak ko measure karti hai.

Subscripts $i$ aur $f$ simply matlab hai initial (pehle) aur final (baad mein). Koi maths nahi — labels hain.

6. Reduced mass — $μ$

Ise invent kyun karein? KE-loss formula tidy $Δ K E = \frac{1}{2} μ u_{r e l}^{2}$ mein collapse ho jaati hai — ordinary $\frac{1}{2} m v^{2}$ jaise hi shape, lekin do bodies ki relative motion ke liye. Ye two-body problem ko ek one-body picture mein package kar deta hai. Tum ise Reduced mass mein dobara miloge.

7. Relative velocity — $u_{r e l} = u_{1} - u_{2}$

Picture. Object 2 par sawar ho jaao. Object 1 tumhari taraf $u_{1} - u_{2}$ par rush karta hai. Jab wo chipak jaate hain, wo closing motion bilkul destroy ho jaati hai — chipakne ke baad, koi bhi ek doosre ke relative nahi hilta. Wahi destroyed relative-motion energy hai $Δ K E$ .

Topic ko iski zaroorat kyun hai. $Δ K E$ sirf is relative velocity par depend karti hai, squared. Ye directly Coefficient of restitution se bhi link karti hai (jo yahan $0$ hai — bilkul bounce-apart nahi).

8. Centre of mass — $v_{c m}$

Poore topic ka punchline: shared final velocity $v$ bilkul $v_{c m}$ ke identical hai. Ek saath chipakna simply matlab hai dono bodies centre of mass ke relative rest mein aa jaati hain, saari relative-motion KE khatam ho jaati hai. Dekho Centre of mass motion.

Foundations topic ko kaise feed karte hain

Ise is tarah padho: mass aur signed velocity momentum aur kinetic energy build karte hain; conserved momentum shared final velocity deta hai (= COM velocity); reduced mass aur relative velocity wo KE deti hain jo khatam hoti hai — aur saath mein explain karte hain kyun loss maximal hai.

Equipment checklist

Khud ko test karo — right side cover karo. Agar koi bhi answer fuzzy lage, parent note se pehle us section ko dobara padho.

m

kya measure karta hai, ek phrase mein?

Stuff ki quantity / motion mein change ki resistance (kg).

Velocity ko

+

-

sign kyun carry karna chahiye?

Ye speed hai direction ke saath; leftward motion ek rightward-positive line par negative hoti hai.

Momentum kya hai aur uska formula?

"Rokna kitna mushkil hai"

= p = m v

; ye

v

ka sign rakhta hai.

Crash mein energy nahi, momentum kyun conserved hota hai?

Internal collision forces equal aur opposite hote hain, isliye koi baahri force na hone par total momentum unchanged rehta hai; energy heat mein leak ho sakti hai.

Kinetic-energy formula likho aur batao

v

kyun square hota hai.

K E = \frac{1}{2} m v^{2}

;

v^{2}

work–energy theorem se aata hai aur KE ko ek curve ki tarah badhata hai, hamesha positive.

Δ

ka kya matlab hai?

"Mein change" — yahan

Δ K E = K E_{i} - K E_{f}

, woh energy jo lost hui.

Reduced mass define karo aur equal masses

m

ke liye iski value do.

μ = \frac{m _{1} m _{2}}{m _{1} + m _{2}}

; equal masses ke liye

m /2

hoti hai.

u_{r e l}

kya hai aur jab bodies chipakti hain toh uska kya hota hai?

u_{r e l} = u_{1} - u_{2}

, closing speed; chipakna ise poori tarah destroy kar deta hai.

Shared final velocity kis ke equal hoti hai?

Centre-of-mass velocity

v_{c m} = \frac{m _{1} u _{1} + m _{2} u _{2}}{m _{1} + m _{2}}

Ek sentence mein, straight-line momentum vs curved energy key kyun hai?

Tum momentum-sum ko fixed rakhte hue energy-sum ko lower kar sakte ho, aur chipakna us dip ke neeche hit karta hai — maximum KE loss.

Connections

Conservation of Linear Momentum — $p = m v$ se bana wo surviving law.
Work–Energy theorem — jahan se $\frac{1}{2} m v^{2}$ aur uska $v^{2}$ aata hai.
Reduced mass — yahan introduce ki gayi $μ$ packaging.
Centre of mass motion — shared $v$ hi $v_{c m}$ hai.
Coefficient of restitution — chipakna matlab $e = 0$ .
Elastic collisions — opposite extreme jahan KE poori tarah survive karti hai.
Ballistic pendulum — in foundations ka ek classic application.