1.1.18 · D1Graphs — x-t, v-t, a-t; areas and slopes meaning
Ek motion graph ek tasveer hai jisme dikhta hai ki ek number time ke saath kaise badalta hai . Poora topic sirf do tareekon se us tasveer ko padhna hai: steepness (slope) batata hai abhi rate of change kya hai , aur neeche ki jagah (area) batata hai ab tak kitna pile up hua hai .
Pehle hum un graphs ko padhna seekhein, usse pehle humein har woh symbol earn karna hoga jo parent note mein diya gaya hai. Neeche, har cheez bilkul scratch se banayi gayi hai: plain words → picture → topic ko yeh kyun chahiye . Upar se neeche padho; har block uske upar wale pe tika hua hai.
hota kya hai
Graph ek flat sheet hai jisme do number-lines right angles pe ek doosre ko cross karti hain.
Horizontal line (drawing ki x-axis kehte hain) hum hamesha time ke liye use karenge.Vertical line (drawing ki y-axis ) hum jo bhi track kar rahe hain uske liye use karte hain — position, ya velocity, ya acceleration.
Sheet pe ek point ek pair hota hai "(yeh time, woh value)". Har instant ke liye ek point banao aur woh milke ek curve ban jaate hain — motion ki kahani .
Intuition Picture kyun banate hain?Table mein numbers motion ki shape chhupa dete hain. Ek curve se aapki aankhein turant dekh leti hain "yahan seedha upar ja raha hai", "wahan flat hai", "wapas neeche aa raha hai". Topic isliye exist karta hai kyunki us shape ki do visual features — steepness aur enclosed space — exactly wahi physics nikali hain jo humein chahiye.
t t tee ) time hai, seconds (s) mein measure hota hai. Is topic ke har graph mein t
Δ Δ delta , ek triangle) padha jaata hai "change in" . Yeh koi number nahi jo kuch multiply kare — yeh ek prefix hai jiska matlab hai "final minus initial".
Δ t = t final − t initial , Δ x = x final − x initial Picture: Δ t Δ x
Δ x Δ x Kyun sahi lagta hai: do symbols side by side hone par usually multiply karte hain.
Fix: Δ x
x ( t ) x position hai: ek chosen starting mark se kitni door hai, metres (m) mein measure hoti hai. Uske baad likha chhota "( t ) x of t time pe depend karti hai — mujhe ek time do aur main tumhe position wapas dunga."
Picture: ek ruler pe sliding bead; x x height hai.
Position negative ho sakti hai — iska matlab sirf "starting mark ke doosri taraf" hai. Yahan sign direction batata hai, ek baat jis par hum lagaataar rely karte hain. Dekho Vectors — Components and Signs aur Distance vs Displacement .
Ab hamari do badi readings mein se pehli.
Definition Slope — "rise over run"Curve pe do points chuno. Left wale se right wale ki taraf jao. Run hai kitna right gaye (Δ t rise hai kitna upar chadhhe (Δ x slope = run rise = Δ t Δ x . Picture: pahaad ki steepness. Gentle ramp = chhota slope; cliff = bahut bada slope; downhill = negative slope; flat = zero slope.
Intuition Yeh tool kyun — slope kyun, kuch aur kyun nahi?Hum jawaab chahte hain "tracked quantity har second mein kitni tezi se badal rahi hai?" "Har second" ka matlab literally hai "quantity mein change ko time mein change se divide karo" — woh ratio hi slope hai. Koi aur akela number "rate" ko itni directly capture nahi karta. Isliye topic jaise hi "kitni tezi se" poochha jaata hai, slope ki taraf jaata hai.
Upar wala slope do points use karta hai. Lekin motion un do points ke beech mein speed change kar sakta hai. Ek exact instant pe rate paane ke liye hum run ko almost zero tak shrink karte hain.
lim Δ t → 0 lim limit ka short form) neeche "Δ t → 0 dekho ki ratio kya settle hota hai jab do points ek saath slide hote hain. Arrow → Picture: ek stretched chord (do points se line) pivoting karti hai jab tak woh curve ko ek point pe sirf kiss nahi karti — woh final line tangent hai, aur uski steepness instantaneous rate hai.
d t d x Δ t Δ x d t d x dee-x by dee-t ". Yeh derivative hai: tangent ka slope, yaani ek instant pe rate.
d t d x = lim Δ t → 0 Δ t Δ x
Do-point slope aur tangent slope ka fark exactly average aur instantaneous ka fark hai — dekho Instantaneous vs Average Velocity . Δ d Differentiation and Integration in Physics mein hai.
v Velocity v rate of change hai — har second mein kitne metres milte hain. Toh yeh exactly x t v = d t d x . v v = 0
a Acceleration a rate of change hai — aapki speedometer needle kitni tezi se chalti hai. Yeh v t a = d t d v .
Chain kyun?
Har variable uske upar wale ka slope hai: position → (slope) → velocity → (slope) → acceleration. Yeh "slopes slide down " ladder hai jo parent note baar baar repeat karta hai. Teeno ki zaroorat hai kyunki physics ka sawaal "motion kaise change ho raha hai?" har level pe poochha ja sakta hai.
Slope ladder pe neeche jaata hai. Wapas upar jaane ke liye (a → v → x
Definition Curve ke neeche area — patli strips pile karnaCurve aur time axis ke beech wale region ko bahut saari lambi patli rectangles mein katao, har ek ki width d t curve ke neeche ka area hai.
Picture: curve ke neeche ki jagah bharne wali skinny planks ki ek picket fence.
∫ t 1 t 2 d t ∫ S um ke liye) matlab "un sab strips ko jodo". Chhote numbers t 1 t 2 kis time se kis time tak sum karna hai. Toh
∫ t 1 t 2 v d t = ( height v , width d t wali strips ka sum ) = v – t curve ke neeche ka area .
Area "ab tak ka total" kyun batata hai
Ek tiny strip mein, velocity almost change nahi karti, toh covered chhoti distance hai (speed)×(time) = height × width = strip ka area . Saari strips jodo ⇒ total displacement. Isliye "area" accumulated quantities ke liye natural tool hai, jaise "slope" rates ke liye tha. Poora slope↔area partnership Differentiation and Integration in Physics mein hai.
Common mistake "Kisi bhi graph pe koi bhi area meaningful hota hai."Kyun sahi lagta hai: v t Fix: area tabhi meaningful hota hai jab height×width ek useful unit de. x t v t a t Δ v
Definition Kinematic equations meinu v
u initial velocity (value at t = 0 v final velocity (value at time t
Picture: straight v t u v u v Equations of Uniformly Accelerated Motion derive karne ke liye use karta hai. Ek concrete example hai dropped ball, jahan poore time a = − g Free Fall and Projectile Motion mein cover hota hai.
velocity v = slope of x-t
acceleration a = slope of v-t
area = sum of thin strips
area of v-t = displacement
area of a-t = change in v
derive u plus a t and half a t squared
Khud test karo — right side cover karo aur zyabon se jawab do.
Δ t Δ "Time mein change" = final minus initial; Δ prefix hai, kabhi koi number nahi jo multiply kare.
In graphs mein se kisi pe bhi curve ki height kya represent karti hai? Us time pe tracked quantity (x v a
Slope ko words mein? Rise over run = (vertical mein change) ÷ (horizontal mein change) = rate of change .
lim Δ t → 0 Do points ko slide karke paas laata hai jab tak line tangent nahi ban jaati — instantaneous rate deta hai.
d t d x "Dee-x by dee-t" — derivative, x t
∫ t 1 , t 2 Yeh patli strips ka sum hai; t 1 , t 2
v t Har strip ka area = height×width = velocity×tiny-time = tiny distance; summed strips = total displacement.
x t Uski units metre·seconds hain, jo kisi physical quantity se correspond nahi karti.
Constant-a u v u t = 0 v t
Ladder pe "slope" kis taraf jaata hai, aur "area" kis taraf? Slope neeche jaata hai x → v → a upar jaata hai a → v → x
