Coefficient of restitution e = (v₂ − v₁) - (u₁ − u₂): Foundations

1.4.8 · D1 · Physics › Momentum & Collisions › Coefficient of restitution e = (v₂ − v₁) - (u₁ − u₂)

Ye ek foundations page hai. Hum yahan collisions solve nahi karenge — hum ye pakka karenge ki parent topic mein jo bhi letter aur symbol use hote hain, unka matlab tumhe wahan jaane se pehle se pata ho.

1. Ek number line aur ek direction (sabse humble symbol: sign)

Kisi bhi velocity se pehle humein ek tarika chahiye ye batane ka ki koi cheez kis taraf move kar rahi hai. Physics ye number line se karta hai: hum ek direction choose karke use "positive" kehte hain.

Figure — Coefficient of restitution e = (v₂ − v₁) - (u₁ − u₂)

Figure dekho. Red arrow hamari chosen positive direction hai. Velocity ki size batati hai kitna fast; sign batata hai kis taraf. Is poore topic mein har velocity symbol secretly ek sign carry karta hai — sign bhool gaye toh har collision ka answer galat ho jayega.

2. Speed vs velocity, aur letters $u$ aur $v$

Topic do families of letters use karta hai, aur choice deliberate hai:

3. Relative velocity — do velocities ka difference

Ye topic se pehle master karne wala sabse important idea hai. Jab do cheezein ek hi line pe move karti hain, ek doosre ki taraf kitni tezi se close in ho rahi hai?

Figure mein dono cars right move kar rahi hain, lekin peeche wali car (red) faster hai. Velocities subtract karna aisa hai jaise front car pe baith jaao: wahan se front car still lagti hai, aur peeche wali car exactly $u_{1} - u_{2}$ pe tumhari taraf aati hai. Isliye velocities ka difference approach measure karta hai.

4. Approach aur separation (aur indices kyun swap hote hain)

Swapped order of subscripts notice karo, aur figure mein dekho kyun:

Crash hone ke liye, chaser (body 1) ko aage wale (body 2) se faster hona chahiye: $u_{1} > u_{2}$ , toh $u_{1} - u_{2} > 0$ . Use " $1$ minus $2$ " likhne se approach positive rehta hai.
Crash ke baad, unhe alag hona chahiye, yaani front body (body 2) faster ho jaati hai: $v_{2} > v_{1}$ , toh $v_{2} - v_{1} > 0$ . Use " $2$ minus $1$ " likhne se separation positive rehta hai.

Ab hum wo symbol assemble kar sakte hain jiske naam pe poora topic hai. Coefficient of restitution bas separation divided by approach hai:

5. Mass, momentum, aur ek equation kyun kaafi nahi hai

Is ek equation mein crash ke baad do unknowns hain: $v_{1}$ aur $v_{2}$ . Ek equation, do unknowns — solve nahi ho sakta. Woh gap exactly wahi hole hai jo coefficient of restitution fill karta hai: ye doosri equation hai. Pehli equation ki poori story ke liye Conservation of Linear Momentum dekho, aur $e$ ke deeper "impulse ratio" meaning ke liye Impulse and Momentum dekho.

6. Squares, roots, aur $h_{2} / h_{1}$ shortcut

Bouncing-ball formula mein square root () use hoti hai. Yahan zero se jaante hain kyun.

Bouncing ball ke liye hum do heights track karte hain, aur matter karta hai kaun si kaun si hai:

Drop-height $h$ aur impact speed $v$ $h = \frac{v ^{2}}{2 g}$ se linked hain, jahan $g \approx 9.8 m/s^{2}$ gravity ka downward pull hai. Height speed squared pe depend karta hai. Toh heights ka ratio speeds squared ka ratio hai: $\frac{h _{2}}{h _{1}} = (\frac{v _{rebound}}{v _{impact}})^{2} = e^{2} \Rightarrow e = \frac{h _{2}}{h _{1}} .$

Sab kuch topic mein kaise jaata hai

Neeche ki picture "prerequisite map" hai: arrows follow karo aur is page ka har foundation ek akele symbol $e$ mein flow karta hai.

Top se bottom padho: signs hume velocities likhne dete hain; velocities hume differences lene dete hain (relative velocity); differences approach aur separation dete hain, jinका ratio hi $e$ hai. Alag se, mass momentum deta hai, jiska conservation pehli equation hai jo $e$ complete karta hai. Squares aur roots speed-ratio $e$ ko measurable height-ratio mein convert karte hain.

Jab ye pieces click kar jaayein, parent topic aur special cases Elastic Collisions ( $e = 1$ ) aur Perfectly Inelastic Collisions ( $e = 0$ ) pe jaao; energy side Kinetic Energy Loss in Collisions mein hai.

Equipment checklist

Right side cover karo aur har cheez zyaban se jawab do.

Velocity ke aage

+

-

kya batata hai?

Motion ki direction chosen positive direction ke relative (size = kitna fast, sign = kis taraf)

Speed aur velocity mein fark?

Speed ek plain positive size hai; velocity speed hai with a sign (direction)

u

aur

v

kya stand karte hain, aur subscript ka matlab?

u

= velocity before,

v

= velocity after; subscript (

1

2

) batata hai kaun sa body

u_{1} - u_{2}

physically kya represent karta hai?

Approach (closing) speed — impact se pehle body 1 body 2 pe kitna gain karta hai

Coefficient of restitution formula mein likho.

e = \frac{v _{2} - v _{1}}{u _{1} - u _{2}}

= separation speed ÷ approach speed

Separation

v_{2} - v_{1}

(swapped) kyun likha hai,

v_{1} - v_{2}

nahi?

Taaki after-collision separation positive aaye (body 2 aage aur faster ho jaati hai)

e

ki normal range kya hai, aur ye 1 se zyada kab ho sakta hai?

Normal collisions ke liye

0 \leq e \leq 1

;

e > 1

(super-elastic) tab hi jab collision internal energy release kare

Do-body collision ke liye momentum-conservation equation likho.

m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}

e

ko doosri equation ki zaroorat kyun hai?

Momentum ek equation deta hai lekin collision mein do unknown final velocities hain (

v_{1}, v_{2}

)

Bounce formula mein

h_{1}

aur

h_{2}

kya hain?

h_{1}

= initial drop height,

h_{2}

= rebound height

Bounce formula

e = h_{2} / h_{1}

kyun hai,

h_{2} / h_{1}

nahi?

Kyunki height

\propto

speed squared (

h = v^{2} /2 g

), toh height ratio

e^{2}

ke barabar hai