Power — average and instantaneous, units: Question bank

1.3.9 · D5 · Physics › Work, Energy & Power › Power — average and instantaneous, units

Shuru karne se pehle, teen tools ka ek reminder jinhein ye traps exploit karte hain, taaki koi symbol unexplained na rahe:

Recall Teen ideas jinpe neeche har trap lean karta hai

Average power $P_{avg} = \frac{W}{Δ t}$ — total work $W$ (joules mein, J) poore time $Δ t$ (seconds mein, s) mein barabar baanta hua. Ise padho jaise "kaam kiya, poore wait pe spread kiya."
Instantaneous power $P = F \cdot v = F v cos θ$ — abhi is waqt ki rate. Yahan $F$ force hai, $v$ velocity hai, aur $θ$ dono ke beech ka angle hai. Dot $\cdot$ matlab dot product hai: yeh sirf force ka woh hissa rakhta hai jo motion ke along hota hai.
$cos θ$ ek switch hai. $θ = 0^{\circ}$ (motion ke saath push) deta hai $cos θ = 1$ ; $θ = 9 0^{\circ}$ (sideways push) deta hai $cos θ = 0$ ; $θ = 18 0^{\circ}$ (motion ke against push) deta hai $cos θ = - 1$ . Yahi ek factor decide karta hai ki power positive, zero, ya negative hai.

Figure — Power — average and instantaneous, units

Agle chaar figures chaar groups of traps ko anchor karte hain. Inhe line by line refer karo: amber force arrow, cyan velocity arrow, aur unke overlap ka sign — yahi har trap asal mein test kar raha hai.

True or false — justify

(Figure s02 dekho: crate dono taraf same height tak uthti hai — red "job done" area identical hai; sirf clock alag hai.)

Do cranes identical crates ko same height tak uthhaate hain; jo zyada fast hai woh zyada work karta hai

False — same force through same height matlab same work

W = m g h

(mass

m

, gravity

g

, height

h

, jaise upar define kiya); fast crane same work kam time mein karta hai, isliye uski zyada power hai, zyada work nahi.

Ek 100 W bulb hamesha 60 W bulb se zyada energy use karta hai

False — energy hai power × time. Ek 60 W bulb jo pura din jala rahe woh zyada energy use karega ek 100 W bulb se jo sirf ek second ke liye jale; wattage ek rate hai, total nahi.

Agar instantaneous power zero hai, toh force bhi zero hona chahiye

False — power

F v cos θ

tab bhi zero hogi agar

v = 0

ho (wall ko push karna) ya agar

θ = 9 0^{\circ}

ho (force motion ke perpendicular, jaise circular motion mein tension), chahe force kitni bhi badi ho. Figure s03 ka middle panel dekho.

Negative power physically impossible hai

False — jab

θ > 9 0^{\circ}

hota hai toh force motion ka oppose karti hai (

cos θ < 0

), isliye

P < 0

: energy object se bahar li ja rahi hai, jaise brakes ya drag kinetic energy hata rahe hain. Yeh figure s03 ka right panel hai.

kWh aur kW ek hi tarah ki quantity measure karte hain

False — kW power hai (rate, J/s) jabki kWh hai power × time = energy (J). Ek kharche ki speed hai, doosra kharcha hua total amount hai.

Kisi interval mein average power hamesha us interval ke midpoint pe instantaneous power ki value ke barabar hoti hai

False — yeh sirf tab valid hai jab power time mein linearly vary kare (rest se constant force). General curve ke liye average aur midpoint value alag hoti hain — figure s04 mein dono curves compare karo.

Object ki speed double karne par, driving force constant rakhte hue, power double ho jaati hai

True —

θ

fixed hone par,

P = F v cos θ

v

mein linear hai, isliye double speed matlab double power (isliye cars ko high speed maintain karne ke liye bahut zyada power chahiye).

Circular orbit mein ek satellite ko gravity se power milti hai

False — gravity centre ki taraf point karti hai jabki velocity tangential hoti hai, isliye

θ = 9 0^{\circ}

cos θ = 0

, aur

P = 0

; speed aur kinetic energy constant rehti hai (dekho Kinetic Energy and Work-Energy Theorem, aur figure s03 mein perpendicular case).

Spot the error

(Figure s03 aapka reference hai: power ka sign sirf amber force arrow aur cyan velocity arrow ka overlap hai.)

"Power force times distance hai,

P = F d

Yeh work hai, power nahi. Power hai work per unit time,

P = W /Δ t

, ya instantaneous form mein

P = F v

— force times speed, distance nahi.

"Engine 30 kW deliver karta hai, toh 2 s baad usne 30 kW energy deliver ki hai."

Units match nahi karte: kW power hai, energy hai kW × time. Woh

30 kW \times 2 s = 60 kJ

energy deliver karta hai, "30 kW energy" nahi.

"Constant velocity par car ka net power

F v

hai, isliye kinetic energy badhti rehti hai."

Yahan

F v

mein jo

F

hai woh engine ki driving force hai, net force nahi — engine ki power sach mein

F v

hai, lekin constant speed par drag force equal negative power deliver karti hai, isliye net power (sab forces ka sum) zero hai aur kinetic energy constant rehti hai.

"1 horsepower 1 kilowatt se bada hai kyunki ghode strong hote hain."

Galat ordering:

1 hp \approx 746 W

, jo

1 kW = 1000 W

se kam hai. Naam ka size se koi lena-dena nahi.

"Kyunki

P = F \cdot v

mein dot product use hota hai, isliye power ek vector hai."

Do vectors ka dot product ek scalar (plain number) return karta hai. Power mein magnitude aur sign hote hain lekin koi direction nahi — yeh ek scalar hai.

"Average power nikaalने ke liye bas starting aur ending instantaneous powers ka average kar lo."

Yeh sirf tab valid hai agar power time mein linear ho. Generally, average power hai total work over total time, jo zaruri nahi ki do endpoint values ka mean ho (figure s04).

"Watt ke base units

kg m^{2} s^{- 2}

hain."

Ye joule (energy) ke units hain. Watt ek joule per second hai:

kg m^{2} s^{- 3}

— joule ko ek aur second se divide karo. Yeh cancellation figure s02 (right panel) mein dikhaayi gayi hai.

Why questions

Stationary wall ko push karna zero power kyun deliver karta hai

Kyunki

v = 0

hai, isliye

P = F v cos θ = 0

; koi displacement nahi matlab koi work per second nahi, chahe aap kitna bhi zor lagao (dekho Work — definition and W = F·d cosθ).

Ek car engine ko 100 km/h hold karne ke liye 50 km/h se zyada power kyun chahiye, chahe "steady speed" ho

Drag speed ke saath badhta hai, isliye required force bhi badhti hai, aur

P = F v

mein badi force ko badi speed se multiply hota hai — power speed se kahin zyada tezi se badhti hai.

Hum instantaneous power ko limit

lim_{Δ t \to 0} Δ W /Δ t

se define kyun karte hain, sirf

W /Δ t

ki jagah

Kyunki rate moment to moment change ho sakti hai; interval ko ek instant tak shrink karna rate ko abhi is waqt capture karta hai, na ki ek smeared-out average — same logic jaise instantaneous velocity.

Continuously chalne wali machine ke liye

P = F \cdot v

P = W /Δ t

se zyada useful kyun hai

Yeh force aur speed se seedha kisi bhi instant ki rate deta hai, bina kisi interval par total work measure kiye — engines ke liye ideal hai jinका output speed ke saath vary karta hai.

P = F \cdot v

mein ordinary multiplication ki jagah dot product (aur kuch nahi) kyun aata hai

Sirf force ka woh component jo motion के along hai energy transfer karta hai; dot product

F v cos θ

automatically woh component rakhta hai aur perpendicular part discard kar deta hai — figure s03 (left panel) mein bani projection dekho.

Kisi real machine ki output power input power se zyada kyun nahi ho sakti

Kuch input energy hamesha friction aur heat mein lost hoti hai, isliye efficiency

= P_{out} / P_{in} < 1

; 1 se zyada value kuch nahi se energy create kar degi (dekho Energy conservation and efficiency).

Rest se start hone wale ek body ke liye, constant force ke under, average power final power ka

\frac{1}{2}

kyun hota hai

Force constant hai lekin

v = a t

zero se linearly badhti hai, isliye

P = F v

bhi 0 se

P_{final}

tak linearly badhta hai; 0 se shuru hoti straight line ka average uska midpoint hai,

\frac{1}{2} P_{final}

— yeh figure s04 mein straight line hai.

Edge cases

(Ye har ek force–velocity arrow picture hai; figure s03 teen governing geometries dikhata hai — aligned, perpendicular, opposed.)

Ek body constant velocity pe move kar rahi hai, net force zero hai — net power kya hai

Zero, kyunki net force zero hai isliye

F \cdot v = 0

; individual forces (engine, drag) equal-and-opposite non-zero powers deliver kar sakti hain jo cancel ho jaati hain.

Ek force act karti hai lekin object abhi move nahi hua (

v = 0

t = 0

) — us instant pe instantaneous power

Exactly zero, kyunki

P = F v = F \cdot 0 = 0

; power tabhi non-zero hoti hai jab object speed gain kare, chahe work accumulate ho rahi ho.

Poore motion mein force velocity ke perpendicular hai — total work aur average power

Dono zero:

cos 9 0^{\circ} = 0

throughout, isliye koi work nahi hota aur kisi bhi interval mein average power zero hai (jaise charge par magnetic force, ya circular motion mein centripetal force) — figure s03 ka middle panel.

Ek car brake lagaake ruk jaati hai — braking force dwara deliver ki gayi power ka sign

Poore time negative, kyunki friction velocity ka oppose karti hai (

θ = 18 0^{\circ}

cos θ = - 1

); yeh negative power kinetic energy drain karti hai jab tak

v = 0

na ho jaye, us instant pe power zero ho jaati hai.

Ek ideal pendulum ke ek poore oscillation mein, gravity dwara deliver ki gayi average power kya hai

Zero — gravity descending mein positive work aur ascending mein equal negative work karta hai, isliye ek cycle mein net work zero hai aur cycle par average power zero hai.

Ek ball seedhi upar throw ki gayi hai — poori flight mein gravity dwara deliver ki gayi power ke sign ko track karo

Gravity poore time neeche point karti hai. Rise ke dौरान velocity upar hai, isliye force aur velocity opposed hain (

θ = 18 0^{\circ}

) aur power negative hai (gravity kinetic energy drain karti hai). Highest point par

v = 0

hai, isliye us instant power exactly zero hai. Fall ke dौरान velocity neeche hai, gravity ke aligned (

θ = 0^{\circ}

), isliye power positive hai. Sequence hai: negative → zero → positive.

Ek machine jo "2 kW" rated hai, 0 seconds ke liye on ki gayi — deliver ki gayi energy

Zero, kyunki energy = power × time aur

Δ t = 0

; rating ek capacity hai, aur koi time nahi matlab koi energy transfer nahi.