Kinetic energy — derivation: Worked examples — Hinglish deep dive

1.3.4 · D3 · Physics › Work, Energy & Power › Kinetic energy — derivation

Shuru karne se pehle, teen symbols bilkul clear hone chahiye (hum koi bhi symbol use nahi karte jab tak usse earn na kar lein):

$m$ = mass kilograms mein (kg) — object mein kitna "stuff" hai.
$v$ = speed metres per second mein (m/s) — kitna fast, direction ignore karke (isliye hum isse square kar sakte hain).
$K = \frac{1}{2} m v^{2}$ = kinetic energy joules mein (J) — motion-energy, hamesha $\geq 0$ .

Scenario matrix

Is topic ka har problem in cells mein se ek mein aata hai. Neeche ke examples uss cell ke label ke saath hain jo woh cover karte hain, taaki milke poora grid fill ho jaye.

Cell	Case class	Tricky kya hai	Example
A	Direct plug-in ( $m, v$ diye gaye)	kuch nahi — sanity baseline	Ex 1
B	Speed doubles / triples	$K \propto v^{2}$ , $v$ nahi	Ex 2
C	Speeding up → $Δ K > 0$	positive net work	Ex 3
D	Slowing down → $Δ K < 0$	negative work (friction/braking)	Ex 4
E	Zero speed / degenerate input	$K = 0$ , limiting behaviour	Ex 5
F	Direction reverse ho jaye ( $v$ ka sign)	$v^{2}$ sign khatam kar deta hai — koi ΔK nahi	Ex 6
G	Real-world word problem	words ko symbols mein translate karo; braking distance	Ex 7
H	Exam twist: ek hidden variable ke liye solve karo	rearrange karo, sirf plug in mat karo	Ex 8

$10 kg$ ka crate floor par still baitha hai, phir ek mover usse $3 m/s$ tak laata hai. Shuru mein $K$ kya hai, aur kitna net work ne use move kiya? Forecast: kisi cheez ki kinetic energy kya hoti hai jo move nahi kar rahi?

Step 1. Rest par $v = 0$ , isliye $K_{i} = \frac{1}{2} (10) (0)^{2} = 0 J$ . Yeh step kyun? Degenerate (zero-speed) case kinetic energy ka floor hai — tum aur neeche nahi ja sakte, kyunki $v^{2} \geq 0$ .

Step 2. Final: $K_{f} = \frac{1}{2} (10) (3)^{2} = \frac{1}{2} (10) (9) = 45 J$ . Yeh step kyun? Final speed plug in karo.

Step 3. Net work $= K_{f} - K_{i} = 45 - 0 = 45 J$ . Yeh step kyun? Rest se start karke, saara work kinetic energy ban jaata hai — yeh exactly wahi situation hai jo original derivation ne assume ki thi.

Verify: $K \geq 0$ hamesha, aur yahan $K_{i} = 0$ minimum hai ✓. Rest se start karne par, $W_{net} = K_{f}$ (parent ke Step-4 boxed result se match karta hai).

Ex 6 — Cell F: direction reverse ho jaata hai, lekin $K$ ko parwah nahi

Ex 7 — Cell G: real-world word problem (braking distance)

Worked example Ek

$1000 kg$ ki car $30 m/s$ se skid karke ruk jaati hai. Tyres ek constant friction force of $6000 N$ provide karte hain. Yeh kitni door skid karti hai? Forecast: saari motion-energy friction dwara skid distance par "khaayi" jaani chahiye — guess karo: 50 m ya 75 m?

Step 1. Remove ki jaane wali kinetic energy: $K = \frac{1}{2} (1000) (30)^{2} = \frac{1}{2} (1000) (900) = 450, 000 J$ . Yeh step kyun? Car ko rokne ke liye, friction ko $K$ ke size ke barabar negative work karna hoga.

Step 2. Distance $s$ par friction ka work: $W = - F s = - 6000 s$ (negative, kyunki yeh motion ka oppose karta hai). Yeh step kyun? Work = force × distance; friction peeche ki taraf point karta hai, isliye sign negative hai — yeh Work — definition and dot product hai jahan force aur displacement anti-parallel hain.

Step 3. Work–energy theorem: $W = K_{f} - K_{i} = 0 - 450, 000$ . Isliye $- 6000 s = - 450, 000 \Rightarrow s = \frac{450 , 000}{6000} = 75 m .$ Yeh step kyun? Remove ki gayi energy ko car ki energy ke barabar set karo, phir unknown distance ke liye solve karo.

Verify: $6000 \times 75 = 450, 000 J$ ✓. Aur kyunki $K \propto v^{2}$ , $60 m/s$ ki car ko $4 \times 75 = 300 m$ chahiye hogi — yahi wajah hai ki speed limits itni matter karti hain.

Ex 8 — Cell H: exam twist (hidden speed ke liye solve karo)

Recall Main kis cell mein hoon? (self-test)

Sirf $m$ aur $v$ diye gaye hain, kaunsa formula? ::: $K = \frac{1}{2} m v^{2}$ (Cell A). Start aur end speeds diye gaye hain, work chahiye? ::: $W_{net} = \frac{1}{2} m (v_{f}^{2} - v_{i}^{2})$ (Cells C/D). Velocity ka sign flip ho gaya, same magnitude — kya $K$ badla? ::: Nahi, $Δ K = 0$ (Cell F), kyunki $v^{2}$ sign ignore karta hai. $K$ aur $m$ diye gaye hain, $v$ chahiye? ::: $v = 2 K / m$ (Cell H) — square ko root se undo karo. Friction force ke saath stopping-distance question? ::: $F s = K$ set karo, $s$ ke liye solve karo (Cell G).

Worked examples — Kinetic energy — derivation

Scenario matrix

Worked examples

Ex 1 — Cell A: direct plug-in

Ex 2 — Cell B: speed ko triple karna

Ex 3 — Cell C: speeding up, positive net work

Ex 4 — Cell D: slowing down, negative work

Ex 5 — Cell E: zero / degenerate input

Ex 6 — Cell F: direction reverse ho jaata hai, lekin $K$ ko parwah nahi

Ex 7 — Cell G: real-world word problem (braking distance)

Ex 8 — Cell H: exam twist (hidden speed ke liye solve karo)

Connections