Elastic collisions — 1D - solve for final velocities: Foundations

1.4.5 · D1 · Physics › Momentum & Collisions › Elastic collisions — 1D - solve for final velocities

Pehle aapko parent note (topic here) use karne se pehle, usmein aane waale har symbol mein fluent hona hoga. Yeh page har ek ko bilkul zero se build karta hai, uss order mein jisme ek doosre par depend karte hain. Kuch bhi assume nahi kiya gaya.

0. Stage: ek seedhi line aur ek direction

Sab kuch ek frictionless seedhi track par hota hai. Kyunki yeh ek line hai, motion ke sirf do possible directions hain: ek taraf ya doosri taraf.

Yeh topic ko kyun chahiye: poori derivation $+$ aur $-$ numbers ke saath algebra hai. Agar aapne ek sign chod diya, toh "moving left" secretly "moving right" ban jaata hai aur baad ke har equation aapko galat bataata hai. Number line hi physics hai.

1. Mass — symbols $m_{1}$ aur $m_{2}$

Neeche ke chhote numbers — subscripts — sirf name tags hain:

$m_{1}$ ::: "m-one" padhte hain, object 1 ki mass.
$m_{2}$ ::: "m-two" padhte hain, object 2 ki mass.

Yeh multiplication nahi hain aur powers bhi nahi. Yeh sirf batate hain ki kis object ki baat kar rahe hain.

Yeh topic ko kyun chahiye: ek crash ka outcome poori tarah se do masses ke ratio par depend karta hai. Ek heavy cart ka light cart se takrana bilkul alag behave karta hai do equal carts se. Subscripts humare liye do objects ko algebra ke through alag rakhte hain.

2. Velocity — $u$ (pehle) aur $v$ (baad mein)

Hum do alag letters use karte hain taaki hum kabhi "before" aur "after" ko confuse na karein:

Formula Char velocity symbols

Symbol	Padhte hain	Matlab
$u_{1}$	u-one	object 1 ki velocity pehle
$u_{2}$	u-two	object 2 ki velocity pehle
$v_{1}$	v-one	object 1 ki velocity baad mein
$v_{2}$	v-two	object 2 ki velocity baad mein

Memory hook: $u$ alphabet mein $v$ se pehle aata hai, bilkul jaisa before time mein after se pehle aata hai.

Yeh topic ko kyun chahiye: poora problem hai "diye gaye $u_{1}, u_{2}$ , find karo $v_{1}, v_{2}$ ." Yeh char symbols hi question aur answer hain.

3. Momentum — symbol $p = m v$

Ab hum mass aur velocity ko ek quantity mein combine karte hain.

Yeh topic ko kyun chahiye — aur kyun yeh tool, sirf "speed" nahi: hum ek aisi quantity chahte hain jo collision mein conserved ho (constant rahe). Plain speed conserved nahi hoti. Momentum hoti hai — kyunki do carts ek doosre ko jo pushes dete hain woh equal aur opposite hote hain (Newton's third law), toh jo ek gain karta hai doosra lose karta hai, aur total kabhi nahi badalti. Poori "why" ke liye Conservation of Momentum dekho. Yeh humein do mein se Equation 1 deta hai: $m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}$ Ise aise padho: (total momentum before) = (total momentum after).

4. Kinetic energy — symbol $\frac{1}{2} m v^{2}$

Humein ek doora rule chahiye, aur yeh momentum se alag hona chahiye (do identical rules sirf repeat karte, kuch naya nahi dete).

Do features notice karne wali hain, kyunki yeh algebra ka behavior change karti hain:

Yeh topic ko kyun chahiye — kyun yeh doosra tool: ek "elastic" collision defined hai ek aisi collision ke roop mein jisme koi kinetic energy heat, sound, ya dents mein nahi jaati (woh case jisme jaati hai uske liye Inelastic Collisions — 1D dekho). Toh total KE before = total KE after. Yeh Equation 2 hai: $\frac{1}{2} m_{1} u_{1}^{2} + \frac{1}{2} m_{2} u_{2}^{2} = \frac{1}{2} m_{1} v_{1}^{2} + \frac{1}{2} m_{2} v_{2}^{2}$

Recall Do equations = solvable kyun

Do unknowns ( $v_{1}, v_{2}$ ) ko do independent facts chahiye. Momentum ek deta hai; kinetic energy ek genuinely alag deta hai (ismein squares hain). Do equations, do unknowns ⇒ exactly ek answer. Do unknowns ko kitni independent equations chahiye? ::: Do.

5. Notation jo aapko fluently padhni aani chahiye

Parent note kuch shorthands use karta hai. Unhe ek baar yahan decode karo.

6. Relative velocity — approach aur separation speeds

Yeh topic ko kyun chahiye: poori derivation ki punchline yeh hai ki elastic 1D collisions ke liye, approach speed = separation speed ( $u_{1} - u_{2} = v_{2} - v_{1}$ ). Yeh linear rule ugly squared energy equation ko replace kar deta hai aur solve karna aasaan bana deta hai. Yeh Coefficient of Restitution ka $e = 1$ case bhi hai.

7. Sab kuch kaise fit hota hai

Left side ka har box is page ka ek foundation hai; arrows dikhate hain ki woh final answer ko kaise feed karte hain jo parent note compute karta hai.

Equipment checklist

Khud ko test karo — right side cover karo aur zor se jawab do.

Agar rightward positive hai, toh "moving left at 2 m/s" kaise likhoge?

- 2

(velocities signed hoti hain).

m_{2}

mein subscript ka kya matlab hai?

Ek name tag — "object 2 ki mass", multiplication ya power nahi.

Is topic mein

u

aur

v

mein kya fark hai?

u

= collision se pehle ki velocity,

v

= collision ke baad ki velocity.

Momentum define karo aur formula do.

"Kitni motion hai";

p = m v

(signed).

Collision mein momentum conserved kyun hota hai?

Do objects ek doosre ko equally aur oppositely push karte hain, toh total kabhi nahi badalti.

Kinetic energy define karo aur formula do.

Motion ki energy;

\frac{1}{2} m v^{2}

Kinetic energy direction kyun ignore karti hai?

Yeh

v^{2}

use karti hai, aur squaring sign khatam kar deta hai:

(- v)^{2} = v^{2}

Collision "elastic" kya banata hai?

Koi kinetic energy lost nahi hoti (koi heat/sound/dents nahi); total KE conserved rehti hai.

a^{2} - b^{2}

factor karo.

(a - b) (a + b)

— difference of squares.

"Approach speed" kya hai?

u_{1} - u_{2}

, relative velocity — woh speed jis par objects ek doosre ke paas aate hain.

Humein DO conservation laws kyun chahiye?

Do unknowns hain (

v_{1}, v_{2}

); do independent equations unhe fix kar dete hain.