Equations of motion (SUVAT) — derivations from calculus

1.1.16 · Physics › Measurement, Vectors & Kinematics

First principles: do definitions

Sab kuch inhi do definitions se aata hai (calculus form):

$v = \frac{d s}{d t}, a = \frac{d v}{d t}$

Derivation 1 — velocity equation $v = u + a t$

Derivation 2 — displacement equation $s = u t + \frac{1}{2} a t^{2}$

Derivation 3 — timeless equation $v^{2} = u^{2} + 2 a s$

Derivation 4 — average-velocity equation $s = \frac{1}{2} (u + v) t$

Figure — Equations of motion (SUVAT) — derivations from calculus

Flashcards

Kaun se do definitions se saare SUVAT equations bante hain?

v = \frac{d s}{d t}

aur

a = \frac{d v}{d t}

SUVAT ke peechhe key assumption kya hai?

Acceleration

a

constant hai.

Ek line mein

v = u + a t

derive karo.

d v = a d t

u

v

aur

0

t

tak integrate karo →

v - u = a t

v = u + a t

kaunsa variable omit karta hai?

Displacement

s

s = u t + \frac{1}{2} a t^{2}

kaunsa variable omit karta hai?

Final velocity

v

v^{2} = u^{2} + 2 a s

kaunsa variable omit karta hai?

Time

t

Kaunsa chain-rule trick

t

hataata hai?

a = v \frac{d v}{d s}

, kyunki

\frac{d v}{d t} = \frac{d v}{d s} \frac{d s}{d t}

SUVAT mein average velocity

\frac{u + v}{2}

kyun hoti hai?

Constant

a

ke saath velocity linearly badhti hai, isliye uska mean midpoint hota hai.

Constant

a

ke liye

v

–

t

graph ke neeche displacement ka area kaunsi shape ka hota hai?

Ek trapezium jiska area

\frac{1}{2} (u + v) t

hai.

Agar

a

constant NAHI hai, toh kya karna hoga?

Actual

a (t)

ko integrate karo:

v = u + \int a d t

, etc.

Recall Ek 12-saal ke bachche ko explain karo (Feynman)

Socho tum ek toy car ko film kar rahe ho. Speed batata hai ki screen pe uska dot kitni tezi se move kar raha hai. Agar tum use poore time equally push karo, toh uski speed har second ek jaisi chunk se badhti hai — yahi constant acceleration hai. Yeh jaanne ke liye ki woh kahan end hogi, uske saare chhote chhote steps ko jodo (yeh jodna "integration" hai). Ek baar karo → speed milti hai; dobara karo → position milti hai. SUVAT formulas sirf us jodne ke shortcuts hain jab push kabhi nahi badalti.

Connections

Differentiation and Integration — yahan har derivation ka engine yahi hai.
Velocity-time graphs — slopes se $a$ milta hai, areas se $s$ .
Vectors — $s, u, v, a$ vectors hain; SUVAT har axis pe alag apply hoti hai.
Projectile motion — SUVAT horizontal ( $a = 0$ ) aur vertical ( $a = - g$ ) mein split hoti hai.
Simple Harmonic Motion — example jahan $a$ constant nahi hai, isliye SUVAT fail ho jaati hai.
Free fall and g — special case $a = g$ .

Equations of motion (SUVAT) — derivations from calculus

Paanch quantities (SUVAT)

First principles: do definitions

Derivation 1 — velocity equation $v = u + a t$

Derivation 2 — displacement equation $s = u t + \frac{1}{2} a t^{2}$

Derivation 3 — timeless equation $v^{2} = u^{2} + 2 a s$

Derivation 4 — average-velocity equation $s = \frac{1}{2} (u + v) t$

Worked examples

Common mistakes

Flashcards

Connections

Concept Map