1.1.15 · D1Average acceleration vs instantaneous acceleration
Acceleration yeh batata hai ki velocity arrow kitni tezi se badal raha hai — aur hum yeh sawaal do tareekon se pooch sakte hain: ek time stretch ke upar (average) ya ek frozen instant par (instantaneous). Parent page par jo bhi hai woh bas yahi do sawaal hain, symbols Δ lim d / d t
Parent note ko theek se padhne se pehle, tumhe har ek mark khud ka banana hoga jo woh likhta hai. Neeche, har symbol ko kuch nahin se banaya gaya hai: plain words → ek picture → kyun woh topic uske bina nahi jee sakta. Inhe is tarah order kiya gaya hai ki har ek pehle wale par tikta hai.
Position yeh hai ki cheez kahan hai . Ek dimension mein hum ise ek number line par pin karte hain: ek zero chuno (origin), ek positive direction chuno, aur object ki location ek single number hai jise hum x
Figure 1 dekho. Amber dot ruler par kisi mark par baitha hai. Woh mark hai x x kaunsa side .
Intuition Symbol kyun at all?Hum x time ke saath badlti hai , aur hum ise machinery mein daalna chahte hain (subtraction, slopes, limits). Ek word ko subtract nahin kiya ja sakta; ek number ko kar sakte hain.
Jab motion 2-D ya 3-D hoti hai toh hum coordinates ko ek vector mein bundle karte hain aur r Vectors: addition and subtraction .
t Δ t t clock ki reading hai, seconds (s) mein. Greek capital delta Δ "mein change" — literally final value minus initial value .
Δ t = t f − t i Δ x = x f − x i i f
Δ Δ gap ki length hai. Figure 1 mein, Δ x Δ t Δ hamesha ek subtraction hai — end minus start, kabhi ulta nahin.
Δ vector (ek arrow) hoti hai, Δ final minus initial hai, lekin minus ab ek vector subtraction hai:
Δ r = r f − r i , Δ v = v f − v i r i Vectors: addition and subtraction mein hai; figure 4 ise velocities ke liye action mein dikhata hai.
Δ Δ Kyun sahi lagta hai: Δ t 2 t Fix: Δ Δ t single quantity hai jiska matlab hai "clock kitna aage badha." Tum Δ Δ
Velocity yeh hai ki position kitni tezi se badlti hai, aur kis direction mein . Ek time interval mein yeh position mein vector change hai divided by elapsed time:
v a v g = Δ t Δ r = t f − t i r f − r i v = d r / d t
Figure 2 dekho, ek position–time graph : time neeche chalti hai, position side mein upar, aur object ki history ek curve hai. Do moments ke beech velocity un do points ko jodne wali straight line ki steepness (slope) hai: steep = tezi se chal raha hai; flat = khada hai; neeche dhalta hua = negative direction mein chal raha hai. Yeh bilkul Average velocity vs instantaneous velocity ki kahani hai — parent topic isse ek rung upar baitha hai.
Velocity ki zaroorat acceleration se pehle kyun hai? Kyunki acceleration velocity ka change hai . Tum "velocity kitni tezi se badal rahi hai?" nahin pooch sakte jab tak tum velocity khud measure nahin kar sakte. Position → velocity → acceleration ek ladder hai; tum ise ek rung ek baar mein chadhte ho.
Slope = (vertical cheez kitni badli) ÷ (horizontal cheez kitni badli) = "rise over run." Do points ke beech ek straight line ke liye yeh ek single number hai.
Definition Secant aur tangentSecant ek straight line hai jo ek curve ko do points par kaatti hai — yeh unke beech gap ke paare change ko summarise karti hai. Tangent ek straight line hai jo ek curve ko sirf ek point par touch karti hai , wahan uski direction se match karti hui — yeh us single instant par behaviour capture karti hai.
Is section ke baaki hisse ke liye hum one dimension tak restrict karte hain (ek single line par motion), toh velocity vector v v
Figure 3 dekho. Ek velocity–time graph par:
Cyan chord jo curve par do points ko jodti hai woh ek secant hai. Uski slope Δ t Δ v average acceleration (1-D).Amber line jo curve ko ek point par sirf touch karti hai woh tangent hai. Uski slope = instantaneous acceleration (1-D).
Intuition Do tarah ki lines kyun?Secant ko do points chahiye, toh yeh jawab deta hai "gap ke paare kya hua." Tangent ek point par rehti hai, toh yeh jawab deta hai "yahan, abhi, kya ho raha hai." Yahi ek distinction geometric form mein pura parent topic hai.
==Δ t → 0 lim dekho ki ek quantity kya approach karti hai jab Δ t — bina exactly zero plug in kiye (jo forbidden 0/0
Intuition Limit ki pictureFigure 4 mein, left point ko fixed rakho aur right point ko uski taraf slide karo. Secant swing karti hai, aur jaise gap Δ t secant rotate karti hai jab tak tangent ke saath flat nahin ho jaati . Limit us swinging line ki manzil hai. "Shrinking window par average" "instant par value" mein badal jaata hai.
Δ t = 0 Kyun sahi lagta hai: agar hum ek point par value chahte hain, toh Δ t = 0 Fix: Δ t = 0 Δ t Δ v = 0 0 approach karke zero aur trend padhke, kabhi land nahin karti usse.
Jab upar wala limit har instant ke liye ek single number par settle ho jaata hai, hum ise ek naam aur ek compact symbol dete hain — derivative :
a = d t d v = lim Δ t → 0 Δ t Δ v d t d v v d instantaneous rate of change . Dekho Derivatives as rates of change . Yahi symbol hume Section 3 mein promise ki gayi instant-by-instant velocity likhne deta hai, v = d r / d t
Units: velocity m/s mein hai aur hum s mein time se divide karte hain, toh acceleration metres per second per second mein measure hota hai, likha m/s 2
Intuition Sign ek direction hai, "speeding up ya slowing down" nahinJab tum apni line par ek positive direction choose kar lete ho, acceleration ek sign inherit kar leti hai. Lekin woh sign batata hai ki Δ v taraf point karta hai — nahin ki object speed up kar raha hai. Tumhe ise velocity ke sign se compare karna hoga:
a v same sign → object speeding up hai (jaise v > 0 a > 0 a v opposite signs → object slowing down hai (jaise v > 0 a < 0 a = 0
Worked example Same magnitude, opposite meaningEk car + a = − 3 m/s 2 braking kar rahi hai. Ek car − wohi a = − 3 m/s 2 backwards speed up kar rahi hai. Same number, opposite physical story — kyunki velocity ka sign alag hai. Hamesha a v ke relative padho.
Intuition Arrow optional nahin haiv Δ v = v f − v i vector subtraction hai (Section 2) — tumhe arrows flip aur add karne pad sakte hain, sirf numbers subtract nahin (dekho Vectors: addition and subtraction ). Yahi wajah hai ki constant speed par curve mein ghoomne wali car phir bhi accelerate karti hai: speed (arrow length) fixed hai, lekin arrow ghoomta rehta hai, toh Δ v = 0 Uniform circular motion ka core hai.
Figure 5 yeh dikhata hai: do velocity arrows barabar length ke lekin alag direction ke, aur amber Δ v
Number line and position x
Time t and the change symbol delta
Vectors have size and direction
Velocity = change of position per time
Limit as delta t goes to zero
Derivative = instantaneous rate
Instantaneous acceleration
Symbol Δ "Mein change" — hamesha final minus initial (Δ t = t f − t i
Δ r Δ r = r f − r i — ek vector subtraction (arrows subtract karo, sirf lengths nahin).
Ek position–time graph par, slope kya represent karta hai? Velocity (steepness = position kitni tezi se badlti hai).
Secant line kya hoti hai? Ek straight line jo curve ko do points par kaatti hai; uski slope gap ke paare average rate deti hai.
Tangent line kya hoti hai? Ek straight line jo curve ko ek point par touch karti hai; uski slope wahan instantaneous rate deti hai.
Velocity–time graph par, secant ki slope kya deti hai? Average acceleration Δ v /Δ t
Velocity–time graph par, tangent ki slope kya deti hai? Instantaneous acceleration.
Acceleration ki units kya hain? Metres per second squared, m/s 2
lim Δ t → 0 Dekho ki quantity kya approach karti hai jab gap zero ki taraf shrink hoti hai, bina exactly zero set kiye.
Δ v /Δ t Δ t = 0 Tum 0/0
Plain words mein derivative kya hai? Woh instantaneous rate jis par ek quantity doosri ke per unit mein badlti hai.
Ek object negative acceleration ke saath kab actually speed up kar raha hota hai? Jab uski velocity bhi negative ho — a
a Yeh ek vector hai; velocity direction carry karti hai, toh direction changes bhi acceleration count hote hain.
d 2 r / d t 2 Position ko time ke respect mein do baar differentiate karo (ek counter, power nahin).
Velocity se pehle acceleration kyun seekhna chahiye? Acceleration velocity ka rate of change hai — ladder position → velocity → acceleration.