Impulse-momentum theorem — derivation: Question bank

1.4.2 · D5 · Physics › Momentum & Collisions › Impulse-momentum theorem — derivation

Shuru karne se pehle, teen words already clear hone chahiye — agar koi shaky lage, toh pehle parent note dobara padho:

True or false — justify karo

Neeche har answer explain karta hai kyun, kyunki sirf "true/false" kehne se trap chupi reh jaati hai.

True or false: Ek chhoti force badi force jaisi hi momentum change produce kar sakti hai.

True — agar chhoti force kaafi der tak lage. Impulse hai

\overset{ˉ}{F} Δ t

, toh chhota

\overset{ˉ}{F}

aur bada

Δ t

milake same

Δ p

dete hain.

True or false: Impulse aur momentum ke units alag hote hain.

False — dono

kg\cdotm/s

hain. Kyunki

J = Δ p

aur barabar cheezein same units share karti hain,

N\cdots

kg\cdotm/s

hai.

True or false: Agar kisi object par net impulse zero hai, toh uski velocity unchanged rehti hai.

True — zero net impulse matlab

Δ p = 0

, toh

m v_{f} = m v_{i}

; fixed mass ke saath,

v_{f} = v_{i}

True or false: Impulse hamesha positive hota hai kyunki yeh force se aata hai.

False — impulse ek vector hai aur sign carry karta hai. Motion ko oppose karne wali force (jaise pakadne wala haath) travel ki direction mein negative impulse deti hai.

True or false: Jo ball bounce karke wapas aati hai, usse zyaada impulse milta hai uss ball se jo sirf ruk jaati hai, agar incoming speed same ho.

True — rukne par

Δ p = - m v

hota hai; bounce karne par

Δ p = - 2 m v

hota hai kyunki velocity reverse ho jaati hai, isliye impulse double ho jaata hai.

True or false:

J = F Δ t

impulse ki fundamental definition hai.

False — fundamental definition hai

J = \int F d t

F Δ t

wali form sirf ek special case hai jo tab valid hai jab

F

constant ho.

True or false: Impulse–momentum theorem sirf constant forces ke liye kaam karta hai.

False — yeh

F = d p / d t

ko integrate karke derive hota hai, isliye yeh kisi bhi force ke liye kaam karta hai, chahe kitni bhi wild ho, jab tak tum integral (curve ke neeche area) use karo.

True or false: Same impulse se hit kiya gaya ek bhaari object aur ek halka object same velocity gain karte hain.

False — dono same momentum change

Δ p

gain karte hain, lekin

Δ v = Δ p / m

hai, isliye halka object zyaada speed up karta hai.

Error dhundo

Har statement mein ek flaw hai. Use dhundo aur theek karo.

"Ek ball

5 m/s

par wall se takraati hai aur

5 m/s

par bounce karti hai, toh uska momentum unchanged hai."

Speed unchanged hai lekin velocity reverse ho gayi (

+ 5 \to - 5

), toh

Δ p = - 2 m v \neq = 0

. Momentum ek vector hai; direction matter karta hai.

"Collision ke dauran force

200 N

hai, toh impulse

200 N\cdots

hai."

Impulse force times time hota hai,

\overset{ˉ}{F} Δ t

— sirf

N

N\cdots

nahi milta. Contact duration se multiply karo.

"Kyunki

F = ma

hai, hum

J = ma Δ t

likh sakte hain jab bhi force spike kare."

Agar force spike kare, toh

a

constant nahi hai, isliye

ma Δ t

ek aisa acceleration use karta hai jo kabhi exist hi nahi kiya. Iske bajaaye

J = \int F d t

= curve ke neeche area use karo.

"Impulse = velocity–time graph ke neeche area."

Galat graph. Impulse force–time graph ke neeche ka area hota hai. Velocity–time graph ke neeche ka area displacement hota hai.

"Ek airbag crash ke dauran driver par impulse ko kam karta hai."

Impulse toh lagbhag fixed hota hai — driver ka momentum phir bhi zero tak laana padega. Airbag

Δ t

badhata hai, jo average force

\overset{ˉ}{F} = Δ p /Δ t

ko kam karta hai, impulse ko nahi.

"Ek force

0.1 s

tak kaam karti hai aur

5 N\cdots

impulse deti hai, toh peak force

50 N

thi."

Yeh average force (

50 N

) deta hai, peak nahi. Ek spiky force ka peak uske average se zyaada hota hai jabki same area enclose karta hai.

"Ande ko gently pakadne se impulse change ho jaata hai, aur yeh toota nahi."

Impulse (ande ko rokne ke liye zaroori) dono cases mein same hai. Apne haath peeche le jaane se

Δ t

badh jaata hai, force ko ande ke todne ki limit se neeche le jaata hai.

Why wale questions

Derivation mein

F = m a

ki jagah momentum form

F = d p / d t

kyun aata hai?

Kyunki

F = m a

secretly constant mass assume karta hai;

d p / d t

zyaada fundamental form hai aur cleanly

Δ p

mein integrate ho jaata hai.

Integrate karne se pehle hum Newton's law ko

d t

se multiply kyun karte hain?

Time interval par total effect gather karne ke liye —

F d t = d p

"abhi change ki rate" ko "accumulated push" mein convert karta hai,

t_{1}

t_{2}

tak sum karne ke liye ready.

Landing par ghutne modna impact ko soft kyun karta hai?

Tera momentum change zero tak fixed hai, lekin ghutne modne se stopping time

Δ t

badh jaata hai, toh

\overset{ˉ}{F} = Δ p /Δ t

kam ho jaata hai.

Jab hum collision ke dauran detailed force nahi jaante, toh impulse useful kyun hai?

Kyunki

J = Δ p

hume measurable start aur end velocities se net effect nikalne deta hai, bina kabhi messy

F (t)

jaane.

Kisi bhi impulse problem solve karne se pehle hum

+

direction kyun assign karte hain?

Impulse aur momentum vectors hain; consistent sign convention ensure karta hai ki ek "reversal" (doubling) accidentally "cancellation" jaisi na lage.

Impulse ko "work ka time-analogue" kyun kaha jaata hai?

Work force distance par accumulated hoti hai (

\int F d x

) aur energy change karti hai; impulse force time par accumulated hota hai (

\int F d t

) aur momentum change karta hai — dekho Work–Energy Theorem.

Edge cases

Ek time interval ke dauran object par impulse kya hoga jisme net force exactly zero hai?

Zero — koi force nahi matlab koi accumulated push nahi, toh

Δ p = 0

aur us interval mein momentum conserve hota hai.

Agar kisi object ki speed constant hai lekin direction change ho (circular motion), toh kya net impulse zero hai?

Nahi — velocity ek vector hai, toh direction change matlab

Δ v \neq = 0

hai, isliye

Δ p \neq = 0

hai aur ek real net impulse act kiya.

Ek infinitely hard collision ek instantaneous (

Δ t \to 0

) stop degi. Average force ka kya hoga?

\overset{ˉ}{F} = Δ p /Δ t \to \infty

— fixed

Δ p

aur

Δ t

zero tak shrink hone ke saath, force blow up kar jaati hai. Isliye rigid impacts itne damaging hote hain.

Ek force act karti hai, phir reverse ho jaati hai aur interval par equally opposite direction mein act karti hai. Net impulse kya hai?

Zero — force–time graph par positive aur negative areas cancel ho jaate hain, toh

Δ p = 0

hota hai, chahe force poore time present rahi ho.

Agar do objects ek doosre par forces exert karte hain (ek collision), toh unke impulses ki comparison kaise hogi?

Woh equal and opposite hote hain — Newton's third law ke hisaab se forces har instant par equal aur opposite hoti hain, toh unke time-integrals (impulses) bhi hote hain. Yahi Conservation of Momentum ki neenv hai.

Ek ball ko drop kiya aur us speed par pakda jis speed par ek perfect bounce ke saath gayi thi, kya floor se impulse zero hai?

Nahi — velocity reverse hoti hai, toh

Δ p = - 2 m v

hai; floor ko motion flip karne ke liye

2 m v

upward ka impulse dena padta hai.

Connections

Impulse-momentum theorem — derivation — parent derivation jo yeh traps test karti hai.
Newton's Second Law — momentum form jo poore theorem ki neenv hai.
Conservation of Momentum — zero-net-impulse aur equal-and-opposite cases upar.
Force–Time Graphs — "impulse = area" traps yahan rehti hain.
Work–Energy Theorem — "why" section mein contrast kiya gaya distance-analogue.
Collisions — Elastic and Inelastic — bounce-vs-stick impulse doubling action mein.