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

1.1.16 · D1 · Physics › Measurement, Vectors & Kinematics › Equations of motion (SUVAT) — derivations from calculus

Is page pe assume kiya gaya hai ki tumne kuch bhi nahi dekha. Parent note pe koi bhi ek SUVAT formula touch karne se pehle, neeche diye words aur pictures tumhari apni honi chahiye. Hum inhe ek aisi order mein build karte hain jahan har ek cheez sirf pehle wali cheez pe rely karti hai.

1. Position — "abhi yeh kahan hai?"

Ek toy car ki imagine karo jo floor pe rakhe ek lambe ruler pe chal rahi hai. Ek jagah choose karo aur use $0$ bolo. Car ki position bas wahi ruler-mark hai jiske upar wo abhi baith rahi hai — jaise $3\text{m}$.

2. Displacement $s$ — "start se kitna door aur kis direction mein"

Agar car $1\text{m}$ se start kare aur $4\text{m}$ pe end kare, to $s=4-1=3\text{m}$. Agar $4$ se start kare aur $1$ pe end kare, to $s=-3\text{m}$ — same trip length, opposite sign.

3. Time $t$ — "clock ki reading"

Picture: ek stopwatch. Jis instant car apne starting mark pe hai, watch $0$ read karta hai. Baad ke har derivation mein likhte hain "integrate from $t=0$ to $t$" — iska matlab bas yeh hai "stopwatch ke click se abhi tak."

4. Graph — "ek story ki picture"

Do graphs jinpe yeh topic rehta hai:

• position–time ($s$ side pe, $t$ bottom pe): curve batata hai ki car har moment pe kahan thi.
• velocity–time ($v$ side pe, $t$ bottom pe): curve batata hai ki har moment pe wo kitni tez ja rahi thi. Dekho Velocity-time graphs.

Hume graphs chahiye kyunki agle do magic words — slope aur area — graph pe pictures hain, abstract ideas nahi.

5. Slope — "story kitni steep hai"

6. Velocity $u$ aur $v$ — "kitni tez, direction ke saath"

Topic do moments pe same cheez ke liye do naam use karta hai:

• ==$u$== = shuruat mein ($t=0$ pe) velocity — initial velocity.
• ==$v$== = general time $t$ pe velocity — final (current) velocity.

7. Acceleration $a$ — "speed khud kitni tezi se change ho rahi hai"

Velocity–time graph imagine karo. Agar yeh steadily chadhe, to iska slope ek fixed number hai — wahi fixed number $a$ hai. Positive direction mein speed up karna → $a>0$; slow down karna (ya doosre direction mein speed up karna) → $a<0$.

8. Notation $\frac{ds}{dt}$ — "slope, symbols mein likha hua"

To parent ke do starting equations seedhe un pictures se translate hote hain jo humne abhi build ki hain:

Yeh tool kyun, plain arithmetic kyun nahi? Kyunki slope ek instant ke baare mein hai — infinitely thin moment — aur sirf derivative notation "yahan ki steepness, time ke mote chunk pe average nahi" ko capture karta hai. Yeh Differentiation and Integration ka subject hai.

9. Integral $\int$ — "saari choti bits ko jodna" (= area)

10. Signs aur positive direction — "ek direction chuno, uspe tike raho"

Yeh topic ko kaise feed karte hain

Equipment checklist

Khud test karo — tum parent note ke liye tab ready ho jab har reveal tumhare apne answer se match kare.

Connections

• Differentiation and Integration — derivative (slope) aur integral (area) machinery jo is page ne define ki.
• Velocity-time graphs — jahan slope se $a$ milta hai aur area se $s$.
• Vectors — kyun $s,u,v,a$ signs/direction carry karte hain.
• Projectile motion — har axis ke liye sign convention chahiye.
• Simple Harmonic Motion — woh case jahan $a$ constant nahi hota.
• Free fall and g — gravity ke sign ka worked example.