Equations of motion (SUVAT) — derivations from calculus

First principles: the two definitions

Everything flows from these two definitions (calculus form):

$v = \frac{ds}{dt}, \qquad a = \frac{dv}{dt}$

Derivation 1 — velocity equation $v = u + at$

Derivation 2 — displacement equation $s = ut + \tfrac12 at^2$

Derivation 3 — timeless equation $v^2 = u^2 + 2as$

Derivation 4 — average-velocity equation $s = \tfrac12 (u+v)\,t$

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

Flashcards

What two definitions generate all SUVAT equations?

v=\frac{ds}{dt}

and

a=\frac{dv}{dt}

Key assumption behind SUVAT?

Acceleration

a

is constant.

Derive

v=u+at

in one line.

Integrate

dv=a\,dt

from

u

v

and

0

t

→

v-u=at

Which variable does

v=u+at

omit?

Displacement

s

Which variable does

s=ut+\tfrac12at^2

omit?

Final velocity

v

Which variable does

v^2=u^2+2as

omit?

Time

t

What chain-rule trick removes

t

a=v\frac{dv}{ds}

, since

\frac{dv}{dt}=\frac{dv}{ds}\frac{ds}{dt}

Why is average velocity

\tfrac{u+v}{2}

for SUVAT?

Velocity rises linearly with constant

a

, so its mean is the midpoint.

Displacement as area under

v

–

t

graph for constant

a

is what shape?

A trapezium of area

\tfrac12(u+v)t

a

is NOT constant, what must you do?

Integrate the actual

a(t)

v=u+\int a\,dt

, etc.

Recall Explain to a 12-year-old (Feynman)

Imagine you're filming a toy car. Speed tells how fast its dot moves on screen. If you push it equally hard the whole time, its speed goes up by the same chunk every second — that's constant acceleration. To find where it ends up, add up all the tiny steps it took (that adding-up is "integration"). Do that once → you get speed; do it again → you get position. The SUVAT formulas are just shortcuts for that adding-up when the push never changes.

Connections

Differentiation and Integration — the engine behind every derivation here.
Velocity-time graphs — slopes give $a$ , areas give $s$ .
Vectors — $s, u, v, a$ are vectors; SUVAT applies per-axis.
Projectile motion — SUVAT split into horizontal ( $a=0$ ) and vertical ( $a=-g$ ).
Simple Harmonic Motion — example where $a$ is not constant, so SUVAT fails.
Free fall and g — special case $a=g$ .

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, SUVAT equations koi magic formula nahi hain — ye sirf calculus ka seedha result hain jab acceleration constant ho. Do hi basic definitions yaad rakho: velocity matlab position ka rate of change ( $v=ds/dt$ ), aur acceleration matlab velocity ka rate of change ( $a=dv/dt$ ). Bas inko integrate karte jao aur saari equations nikal aati hain.

Pehle $a=dv/dt$ se shuru karo, $dv=a\,dt$ ko integrate karo $u$ se $v$ tak — mil gaya $v=u+at$ . Phir is $v$ ko $ds=v\,dt$ mein daalo aur integrate karo — $s=ut+\tfrac12at^2$ aa gaya. Time ko hatana ho to chain-rule trick lagao: $a=v\,dv/ds$ , isse $v^2=u^2+2as$ nikalta hai. Aur average velocity wali $s=\tfrac12(u+v)t$ to bas $v$ – $t$ graph ke trapezium ka area hai.

Important baat: ye sab sirf tab chalte hain jab $a$ constant ho. Air resistance ho ya SHM ho jahan $a$ badalta hai, wahan ye fail ho jate hain — phir actual $a(t)$ ko integrate karna padta hai. Aur sign ka dhyan rakho: pehle positive direction choose karo, fir gravity ya deceleration ka sign uske hisaab se lagao. Distance aur displacement mat mix karna — $s$ hamesha displacement hai.

Exam tip (80/20): in 4 equations mein se woh choose karo jisme tumhare paas jo variable na ho na chahiye, woh missing ho. Isse seconds mein sahi formula select ho jaata hai.

Equations of motion (SUVAT) — derivations from calculus

The five quantities (SUVAT)

First principles: the two definitions

Derivation 1 — velocity equation $v = u + at$

Derivation 2 — displacement equation $s = ut + \tfrac12 at^2$

Derivation 3 — timeless equation $v^2 = u^2 + 2as$

Derivation 4 — average-velocity equation $s = \tfrac12 (u+v)\,t$

Worked examples

Common mistakes

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Measurement, Vectors & Kinematics

Connections