Power — average and instantaneous, units

1.3.9 · Physics › Work, Energy & Power

WHAT is Power?

Jaise velocity mein hota hai, exactly waise hi do flavours hain:

• Average power — poore time interval par.
• Instantaneous power — kisi ek specific instant par.

Yeh exactly wahi average-vs-instantaneous split hai jo tumne kinematics mein speed ke saath dekha tha. Same idea, alag quantity.

HOW to derive it from first principles

Average power. Time se kyun divide karte hain? Kyunki "rate" ka matlab literally hota hai per unit time. Agar $\Delta t$ time mein $W$ work hota hai:

$\boxed{P_{\text{avg}}=\frac{W}{\Delta t}=\frac{\Delta E}{\Delta t}}$

Instantaneous power. Limit kyun lete hain? Kyunki rate moment to moment change ho sakti hai (accelerate karta car high speed par zyada power deliver karta hai). Interval ko zero tak shrink karo:

$P_{\text{inst}}={\lim }_{\Delta t\to 0}\frac{\Delta W}{\Delta t}=\frac{dW}{dt}$

Power ko force aur velocity se connect karna. Yeh sabse useful form hai. Small displacement $d\vec{s}$ ke liye work ki definition se shuru karo:

$dW=\vec{F}\cdot d\vec{s}$

Dono sides ko $dt$ se divide karo:

$\frac{dW}{dt}=\vec{F}\cdot \frac{d\vec{s}}{dt}$

Lekin $\frac{d\vec{s}}{dt}=\vec{v}$, velocity hai. Toh:

$\boxed{P_{\text{inst}}=\vec{F}\cdot \vec{v}=Fv\cos \theta }$

Units — WHY the watt?

$P=W/\Delta t$ se, SI unit joule per second hai:

$1\text{watt}=1\text{W}=1\frac{\text{J}}{\text{s}}=1\frac{\text{kg}{\text{m}}^2}{{\text{s}}^3}$

(Last form $\text{J}=\text{kg}{\text{m}}^2/{\text{s}}^2$ se aata hai, phir ek aur second se divide karo.)

Doosre units jo tumhe pata hone chahiye:

Worked Examples

Active Recall

Flashcards

Connections

• Work — definition and W = F·d cosθ — power, work per unit time hai.
• Kinetic Energy and Work-Energy Theorem — $\Delta KE$ se total work, average power ko feed karta hai.
• Velocity and instantaneous rate (Kinematics) — same average-vs-instantaneous logic.
• Dot product of vectors — $\vec{F}\cdot \vec{v}$ ke liye zaroori.
• Energy conservation and efficiency — useful vs input power → efficiency.

Concept Map