Foundations — Kinetic energy — derivation

1.3.4 · D1 · Physics › Work, Energy & Power › Kinetic energy — derivation

Pehle se derive karne ke liye tumhe $K=\frac{1}{2}m{v}^2$ aur $W=\vec{F}\cdot \vec{d}$ mein har letter ka pura gyaan hona chahiye. Yeh page har ek ko bilkul scratch se, usi order mein build karta hai jis order mein derivation unhe actually use karti hai. Agar parent page ne assume kiya ki tum koi symbol jaante ho, toh hum usse yahan earn karte hain.

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0. Jo hum bana rahe hain uska map

Isko upar se neeche padho: mass aur acceleration milke force banate hain; force distance ke upar kaam karke work banata hai; kinematics distance ko speed se jodhta hai; ye teeno derivation mein jaate hain; aur bahar nikalti hai $K$. Ab hum har ek box ko unlock karte hain.

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1. Mass $m$ — "kitna samaan hai" wala number

Picture: socho tum ek shopping trolley dhakele raho. Khaali trolley ek chhote se dhakke se aage bhaagg jaati hai; bhari hui trolley usi dhakke se mushkil se hilti hai. Yeh "bharaa-pan" jo tumhare push ko rokta hai wahi mass hai.

Topic ko iska kyun zaroorat hai: kinetic energy poochti hai "kisi chalti cheez mein kitni go-energy hai." Usi speed par ek bhaari cheez zyada energy carry karti hai — isliye $m$ formula ke andar hona chahiye.

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2. Speed $v$ aur velocity — "kitni tez" vs "kitni tez, kis direction mein"

Picture (neeche): ek car ek arrow ke saath. Arrow ki length speed hai; arrow ki direction woh extra information hai jo velocity carry karti hai.

Topic ko iska kyun zaroorat hai: kinetic energy motion ke saath badhti hai. Lekin dhyaan do ki parent $v^2$ use karta hai, $v$ nahi — squaring direction ko phenk deta hai (ek negative velocity ka square positive hota hai). Yeh pehla clue hai ki energy ek scalar hogi. Yeh baat §7 ke liye yaad rakhna.

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3. Acceleration $a$ — speed khud kitni tezi se badlti hai

Picture: car ka pedal dabao aur speedometer ki needle chadhne lagti hai — woh chadhne ki rate acceleration hai. Agar woh har second $2\text{m/s}$ chadhti hai, toh $a=2{\text{m/s}}^2$.

Topic ko iska kyun zaroorat hai: kisi object ko rest se chalana shuru karne ke liye, kuch toh use accelerate karna hoga. Derivation force ki jagah "mass × acceleration" use karti hai, isliye $a$ push se motion tak ka bridge hai.

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4. Force $F$ aur Newton's second law — woh push

Picture (neeche): ek floor par block, ek clear arrow net push $F$ dikhata hua, aur ek dheema arrow resulting acceleration $a$ ko usi direction mein dikhata hua.

Topic ko iska kyun zaroorat hai — aur yahi tool kyun: hum kinetic energy ko motion ($v$) ke terms mein likhna chahte hain, lekin work force ($F$) ke terms mein likhi jaati hai. Newton's law $F=ma$ hi woh ek cheez hai jo force statement ko motion statement mein convert karti hai. Issi liye parent derivation ke Step 2 mein $F=ma$ substitute hota hai. Dekho Newton's Second Law.

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5. Displacement $s$ — push ki direction mein kitni door

Picture: ek dashed track jahan object shuru hua tha (green dot) se jahan khatam hua (red dot) tak; $s$ us seedhe segment ki length hai.

Topic ko iska kyun zaroorat hai: ek force jo kaam kare lekin kuch na hilaye woh zero work karta hai. Energy tabhi transfer hoti hai jab distance tay ki jaaye. Isliye work mein force ko displacement se multiply karna padta hai — jo hume work tak le jaata hai.

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6. Work $W$ aur dot product — distance par force ka kharch

Ab general case, kyunki force aur motion ka same direction mein hona zaroori nahi.

$\cos \theta$ kyun? (figure dekho) Sirf push ka woh component jahan tum ja rahe ho us direction mein actually work karta hai.

• Agar $\theta =0^{\circ }$ (push seedha motion ke along): $\cos 0=1$, toh $W=Fd$ — maximum work.
• Agar $\theta =90^{\circ }$ (push sideways, jaise level chalte hue bag uthana): $\cos 90^{\circ }=0$, toh $W=0$ — koi work nahi.
• Agar $\theta =180^{\circ }$ (push motion ko oppose kare, jaise friction): $\cos 180^{\circ }=-1$, toh $W=-Fd$ — negative work, energy nikal jaati hai. Yahi reason hai ki parent page par Example 3 mein minus sign aata hai.

Topic ko iska kyun zaroorat hai: puri derivation $W=Fs$ se shuru hoti hai (parent Step 1). Dekho Work — definition and dot product.

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7. Scalar vs vector — energy ka koi arrow kyun nahi hota

Picture: velocity ko ek arrow ke roop mein draw kiya jaata hai (§2); kinetic energy ko draw kiya jaata… kuch nahi, bas ek labelled number, kyunki "9 joules pointing north" ka koi matlab nahi.

Topic ko iska kyun zaroorat hai: parent poochta hai kyun $K$ ek scalar hai. Jawab: $K$ ek dot product se bana hai (jo ek number output karta hai) aur $v^2$ se (jo direction khatam kar deta hai). Dono ingredients directionless hain, isliye $K$ bhi directionless hai. Dekho Momentum vs Energy — usme explain hai ki momentum apna arrow kyun rakhta hai jabki energy nahi rakhti.

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8. Squaring, aur $v^2$ (naa ki $v$) kyun aata hai

Topic ko iska kyun zaroorat hai: yeh ek fact hi parent ka headline surprise explain karta hai — speed double karo, energy chaar guni ho jaati hai — aur guarantee karta hai ki $K\ge 0$ hamesha.

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9. Kinematics: $v^2=u^2+2as$ — distance–speed ka bridge

Picture: speed ka ek number line jo $u$ se $v$ tak chadhta hai jab object steady acceleration $a$ ke under distance $s$ slide karta hai. Yeh equation khaas hai kyunki yeh speed aur distance ko time ka zikr kiye bina jodhti hai — yahi derivation chahti hai.

Yahi equation kyun, koi aur kyun nahi? Derivation mein leftover chunk $as$ bachta hai aur usse $v$ ke terms mein chahiye. Kinematic equations mein, $v^2=u^2+2as$ hi akela hai jo $a$, $s$, aur $v$ ko time chhode bina connect karta hai — isliye cleanly substitute ho jaata hai. Dekho Equations of Motion (kinematics).

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10. Symbols ko parent mein wapas daalna

Ab parent ke chaar steps mein har letter earn ho chuka hai:

Symbol	Meaning	Section
$m$	mass (kg)	§1
$v$	final speed (m/s)	§2
$a$	acceleration (m/s²)	§3
$F$	net force (N)	§4
$s$	displacement (m)	§5
$W$	work (J)	§6
$u$	initial speed (m/s)	§9
$K$	kinetic energy (J)	goal

$W\underset{\text{§6}}{=}Fs\underset{\text{§4}}{=}(ma)s\underset{\text{§9}}{=}m\left(\frac{1}{2}{v}^2\right)=\frac{1}{2}m{v}^2=K.$

Ab har equals-sign usi section ki taraf point karta hai jo tumne khud build ki hai. Parent derivation par wapas jao aur tumhe koi unearned symbol nahi milega.

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Equipment checklist

Reveal karne se pehle har ek ka matlab batao — agar koi atka de, toh usi section ko dobara padho.

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Connections

• Work — definition and dot product
• Newton's Second Law
• Equations of Motion (kinematics)
• Work–Energy Theorem
• Momentum vs Energy
• Power — rate of doing work