Foundations — Elastic collisions — 2D - angle relationship

1.4.6 · D1 · Physics › Momentum & Collisions › Elastic collisions — 2D - angle relationship

Pehle tum parent page par 90° result derive karo, har arrow, symbol, aur squiggle ka kuch matlab hona chahiye tumhare liye. Yeh page har ek ko zero se build karta hai. Upar se neeche padho — har block sirf wahi use karta hai jo pehle aaya hai.

1. Ek vector — woh arrow jo direction carry karta hai

Neeche wali figure dekho. Lavender arrow ek velocity hai. Yeh jhuka hua hai, isliye direction hai. Yeh lamba hai, isliye speed badi hai. Do facts, ek arrow.

2. Components — ek arrow ko x-part aur y-part mein kaatna

Figure mein coral arrow ${\vec{v}}_1$ horizontal line ke upar angle ${\theta }_1$ par jhuka hua hai. Ek seedha neeche wali line aur ek flat line daalo — tumhe ek right triangle milta hai:

• flat side (${\theta }_1$ ke adjacent) x-component $v_1\cos {\theta }_1$ hai,
• upright side (${\theta }_1$ ke opposite) y-component $v_1\sin {\theta }_1$ hai.

Signs: components negative ho sakte hain

Formulas $v\cos \theta$ aur $v\sin \theta$ har direction ko already handle karte hain, kyunki $\sin$ aur $\cos$ sign change karte hain depending on ki arrow kahan point karta hai.

3. Vectors jodhna — arrows par "$+$"

Conservation of momentum literally velocity arrows ko add karta hai, isliye humein yeh nail down karna chahiye ki arrows ke liye "$+$" ka matlab kya hai.

4. Identity ${\sin }^2\theta +{\cos }^2\theta =1$

Yeh sach kyun hai? Do components ek right triangle ki legs hain jiska hypotenuse length $v_1$ ka arrow hai. Pythagoras kehta hai leg² + leg² = hypotenuse²: $(v_1\cos {\theta }_1{)}^2+(v_1\sin {\theta }_1{)}^2=v_1^2.$ Dono sides ko $v_1^2$ se divide karo aur tumhe identity milti hai. (Yeh har quadrant mein sach rehta hai: squaring minus signs ko khatam kar deta hai.) Parent page is identity ko Step 5 mein use karta hai yeh check karne ke liye ki $v_1^2+v_2^2=u_1^2$ — woh check yahi identity hai.

5. Velocity symbols jo tumse milenge

Parent page kaafi velocity letters use karta hai. Unke meanings abhi fix karo taaki koi subscript tumhe surprise na kare.

6. Momentum — "quantity of motion" arrow

Kyunki momentum ek vector hai, "total momentum unchanged" actually do promises hai (§3 ke addition rule ko use karke): total right-part unchanged hai aur total up-part unchanged hai. Wahan se parent ki do equations (x aur y) aati hain.

7. Kinetic energy — single "motion-cash" number

8. Dot product — woh tool jo measure karta hai "kitna aligned"

Yeh parent proof ka star tool hai, isliye hum ise carefully build karte hain.

9. Special angles jo tum actually plug in karoge

$37^{\circ }/53^{\circ }$ pair ek physics-classroom convenience hai: real values hain $\sin 37^{\circ }=0.6018\dots$, isliye $0.6$ aur $0.8$ approximations hain jo tidy $3$–$4$–$5$ triangle ke liye choose ki gayi hain. Inhe arithmetic ke liye use karo, lekin jaano ki yeh lagbhag $0.3\%$ rounding error carry karte hain. Parent ka worked example yeh pair use karta hai — dhyan karo yeh $90^{\circ }$ mein add hote hain, aur unke sine/cosine values swap ho jaate hain (right-angle rule ka direct consequence).

Foundations topic ko kaise feed karte hain

Neeche wala map ek dependency chart hai: ek arrow "$A\to B$" padhte hain jaise "tumhe $A$ chahiye pehle $B$ sense banane se." Upar se shuru karo (raw vector), arrows neeche follow karo, aur har path eventually 90 degree angle result mein funnel ho jaata hai neeche. Agar diagram tumhare device par render nahi hota, toh wahi order hai: vector → components → (sign/quadrant rules, addition, sin/cos identity); vector → momentum → conservation → two equations; kinetic energy → elastic → one energy equation; vector → dot product → "dot = 0 means 90°". Woh sab final result par milte hain.

Equipment checklist

Khud ko test karo — tum parent page ke liye ready ho jab tum har ek ka jawab de sako: