Power — average and instantaneous, units

WHAT is Power?

There are two flavours, just like with velocity:

Average power — over a whole time interval.
Instantaneous power — at one specific instant.

This is exactly the same average-vs-instantaneous split you saw with speed in kinematics. Same idea, different quantity.

HOW to derive it from first principles

Average power. WHY divide by time? Because "rate" literally means per unit time. If work $W$ is done in time $\Delta t$ :

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

Instantaneous power. WHY take a limit? Because the rate can change moment to moment (a car accelerating delivers more power at high speed). Shrink the interval to zero:

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

Connecting power to force and velocity. This is the most useful form. Start from the definition of work for a small displacement $d\vec{s}$ :

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

Divide both sides by $dt$ :

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

But $\dfrac{d\vec{s}}{dt} = \vec{v}$ , the velocity. So:

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

Units — WHY the watt?

From $P = W/\Delta t$ , the SI unit is joule per second:

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

(The last form comes from $\text{J} = \text{kg}\,\text{m}^2/\text{s}^2$ , then divide by another second.)

Other units you must know:

Unit	Value	Note
1 W	1 J/s	SI base
1 kW	$10^3$ W
1 MW	$10^6$ W	power plants
1 horsepower (hp)	$\approx 746$ W	engines
1 kWh	$3.6\times10^6$ J	unit of ENERGY, not power!

Figure — Power — average and instantaneous, units

Worked Examples

Worked example Example 3 — Power varies with time

A particle of mass $2$ kg starts from rest with constant acceleration $a = 3\text{ m/s}^2$ . Find power at $t = 4$ s, and average power over $0$ – $4$ s.

Force: $F = ma = 2\times 3 = 6$ N (constant). Velocity at $t=4$ : $v = at = 3\times 4 = 12$ m/s. $P_{\text{inst}}(4) = Fv = 6\times 12 = 72\text{ W}.$ Why this step? Even though force is constant, $v$ grows with time, so power grows — instantaneous power is not constant here.

For average power, compute total work via KE: $v^2 = (12)^2$ , $\;W = \tfrac12 m v^2 = \tfrac12 (2)(144) = 144$ J. $P_{\text{avg}} = \frac{144}{4} = 36\text{ W}.$ Notice $P_{\text{avg}} = 36 = \tfrac12 \times 72 = \tfrac12 P_{\text{final}}$ — makes sense: power rose linearly from 0 to 72, so its average is halfway.

Active Recall

Recall Quick self-test (cover the answers)

Q: What is the SI unit of power in base units? → $\text{kg m}^2\text{ s}^{-3}$
Q: Why is power zero when you push a stationary wall? → $v=0 \Rightarrow P=Fv=0$ ; no energy transferred per second.
Q: When does $P_{\text{inst}} = P_{\text{avg}}$ ? → When power is constant over the interval (e.g. constant force AND constant velocity).
Q: Is kWh power or energy? → Energy.

Recall Feynman: explain to a 12-year-old

Imagine two kids each carry 10 bricks up the stairs. They both do the same job. But one runs up in 5 seconds and the other strolls up in 50 seconds. The runner is more powerful — he uses his energy faster. Power is just "how quickly you spend your energy." The unit "watt" tells you how many joules of energy you burn every single second. A 60-watt bulb eats 60 joules each second!

Flashcards

Define power

The rate at which work is done or energy is transferred with respect to time.

Average power formula

P_{avg} = W/\Delta t = \Delta E/\Delta t

Instantaneous power formula (calculus)

P = dW/dt

Power in terms of force and velocity

P = \vec F\cdot\vec v = Fv\cos\theta

Derive

P=\vec F\cdot\vec v

dW=\vec F\cdot d\vec s

, divide by

dt

dW/dt = \vec F\cdot(d\vec s/dt) = \vec F\cdot\vec v

SI unit of power

watt (W) = J/s = kg·m²/s³.

1 horsepower in watts

≈ 746 W.

1 kWh in joules

3.6\times10^6

Is kWh power or energy?

Energy (power × time).

Why is power zero when pushing a wall hard?

v=0

, so

P=Fv=0

; no work done per unit time.

Car at constant 20 m/s with 1500 N drive force, power?

P=Fv=30000

W = 30 kW.

When does instantaneous = average power?

When power stays constant over the interval.

Relation of average to peak power for constant force from rest

P_{avg}=\tfrac12 P_{final}

(power rises linearly).

Connections

Work — definition and W = F·d cosθ — power is work per unit time.
Kinetic Energy and Work-Energy Theorem — total work via $\Delta KE$ feeds average power.
Velocity and instantaneous rate (Kinematics) — same average-vs-instantaneous logic.
Dot product of vectors — needed for $\vec F\cdot\vec v$ .
Energy conservation and efficiency — useful vs input power → efficiency.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, power ka matlab hai "kaam kitni JALDI ho raha hai" — yani rate of doing work. Do crane same crate ko same height tak utha sakti hain (same work, same energy), par jo crane 2 second mein utha de woh zyada powerful hai. Isliye power = work / time. Yeh bilkul speed jaisa concept hai: jaise speed = distance/time hoti hai, waise hi power = work/time.

Do versions hain. Average power poore interval ke liye: $P_{avg}=W/\Delta t$ . Instantaneous power ek hi pal ke liye: $P=dW/dt$ . Ek mast formula yaad rakho — $P=\vec F\cdot\vec v=Fv\cos\theta$ . Iska matlab: power depend karti hai ki tum kitna zor laga rahe ho aur cheez kitni tez chal rahi hai. Agar deewar ko dhakka do par woh hile hi nahi ( $v=0$ ), to power zero — chahe jitna zor lagao, koi energy transfer nahi ho rahi per second.

Unit SI mein watt hai, jo $1$ J/s ke barabar hai (base units mein $\text{kg m}^2/\text{s}^3$ ). 1 horsepower $\approx 746$ W hota hai. Aur sabse important confusion: kWh power nahi, ENERGY hai! kWh $=$ kilowatt $\times$ hour $=$ power $\times$ time $= 3.6\times10^6$ J. Bijli ka bill isi kWh (energy) ka aata hai. Exam mein yeh trap bahut aata hai, isliye yaad rakho: jis cheez mein "per second" ya rate ka concept ho woh power, aur jo total spend ho woh energy.

Power — average and instantaneous, units

WHAT is Power?

HOW to derive it from first principles

Units — WHY the watt?

Worked Examples

Active Recall

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Work, Energy & Power

Connections