Equations of Motion - Class 9 Science

Motion Ke Teen Jaadui Formulas ✨

Jab kisi object ka acceleration uniform (constant) ho, tab teen equations se hum kuch bhi nikaal sakte hain - velocity, distance, time, acceleration. Ye hi Equations of Motion hain, aur poore physics mein sabse zyada use hone wale formulas mein se hain.

📌 Symbols yaad rakho:
u = initial velocity (shuruaat ki velocity, m/s)
v = final velocity (aakhir ki velocity, m/s)
a = acceleration (m/s²)
s = distance/displacement (m)
t = time (s)

📐 Teeno Equations

1️⃣ Pehli Equation (velocity-time)

v = u + a t

Ye batati hai ki t samay baad velocity kya hogi. Isme distance (s) nahi hai.

2️⃣ Doosri Equation (position-time)

s = u t + ½ a t²

Ye batati hai ki t samay mein kitni doori tay hogi. Isme final velocity (v) nahi hai.

3️⃣ Teesri Equation (velocity-position)

v² = u² + 2 a s

Ye velocity aur distance ko jodti hai - isme time (t) nahi hai! Jab question mein time na diya ho, ye equation use karo.

📌 Kaunsi equation kab? (Sabse important tip)
▸ Time diya hai, distance nahi chahiye → v = u + at
▸ Time diya hai, distance chahiye → s = ut + ½at²
▸ Time nahi diya (na hi poochha) → v² = u² + 2as

🔍 Derivation (Graph Se)

Ye teeno equations velocity-time graph se nikalti hain (pichla lesson yaad karo):

✔ Pehli equation:

v-t graph ka slope = acceleration.
a = (v − u) ÷ t → at = v − u → v = u + at ✔

✔ Doosri equation:

v-t graph ke neeche ka area = distance. Shape ek trapezium hai:
s = ½ × (u + v) × t
Ab v ki jagah (u + at) rakho:
s = ½ × (u + u + at) × t = ½ × (2u + at) × t
s = ut + ½at² ✔

✔ Teesri equation:

s = ½(u + v)t, aur pehli equation se t = (v − u) ÷ a
s = ½ (u + v) × (v − u) ÷ a = (v² − u²) ÷ 2a
2as = v² − u² → v² = u² + 2as ✔

🧮 Solved Numericals (Step-by-Step)

✔ Example 1 - Rest se shuru:

Ek car rest se chalti hai aur 4 m/s² se accelerate karti hai. 5 second baad (a) velocity kya hogi? (b) kitni doori tay hogi?

Diya hai: u = 0, a = 4 m/s², t = 5 s
(a) v = u + at = 0 + 4 × 5 = 20 m/s
(b) s = ut + ½at² = 0 + ½ × 4 × 5² = ½ × 4 × 25 = 50 m

✔ Example 2 - Acceleration aur distance:

Ek bus ki velocity 10 m/s se badhkar 4 second mein 30 m/s ho jaati hai. (a) acceleration (b) distance?

Diya hai: u = 10, v = 30, t = 4
(a) v = u + at → 30 = 10 + a(4) → 20 = 4a → a = 5 m/s²
(b) s = ut + ½at² = 10(4) + ½(5)(16) = 40 + 40 = 80 m
Check: s = ½(u+v)t = ½(40)(4) = 80 m ✔ (dono tareeke se same!)

✔ Example 3 - Brake (time nahi diya → teesri equation):

Ek car 20 m/s se chal rahi hai aur 50 m mein ruk jaati hai. Retardation nikalo.

Diya hai: u = 20, v = 0, s = 50, t = ? (nahi diya!)
Isliye v² = u² + 2as use karo:
0² = 20² + 2a(50) → 0 = 400 + 100a → a = −4 m/s²
Matlab retardation = 4 m/s².

✔ Example 4 - Free fall (gravity):

Ek patthar building se gira. 3 second baad (a) velocity (b) kitna neeche gaya? (g = 10 m/s²)

Diya hai: u = 0, a = g = 10, t = 3
(a) v = 0 + 10(3) = 30 m/s
(b) s = 0 + ½(10)(9) = 45 m

✔ Example 5 - Train rukne mein kitna time?

Ek train 72 km/h se chal rahi hai. Brake lagane par 2 m/s² ki retardation hoti hai. (a) rukne mein kitna time? (b) kitni doori tay karegi?

Step 1 - unit badlo: u = 72 × 5/18 = 20 m/s, v = 0, a = −2 m/s²
(a) v = u + at → 0 = 20 + (−2)t → 2t = 20 → t = 10 s
(b) s = ½(u+v)t = ½(20+0)(10) = 100 m

✔ Example 6 - Distance nikalna:

u = 5 m/s, a = 2 m/s². 6 second mein kitni doori?

s = ut + ½at² = 5(6) + ½(2)(36) = 30 + 36 = 66 m

⚠️ Numericals Solve Karne Ke Tips

✔ Pehle sab likho: u, v, a, s, t mein se jo diya hai wo likho, aur jo poochha hai wo mark karo.

✔ Units check karo: km/h ko m/s mein badlo (× 5/18) - warna jawab galat aayega!

✔ Rukne ka matlab v = 0, aur "rest se shuru" ka matlab u = 0.

✔ Retardation mein a negative lagao (jaise a = −2 m/s²).

✔ Time nahi diya? Teesri equation (v² = u² + 2as) use karo.

✏️ Khud Try Karo (Practice)

Q. Solve karo (answers neeche):

u = 0, a = 5 m/s², t = 4 s. v aur s nikalo.
u = 15 m/s, v = 0, t = 3 s. Retardation nikalo.
u = 0, a = 2 m/s², s = 100 m. Final velocity v nikalo.

Answers: 1) v = 0+5(4) = 20 m/s, s = ½(5)(16) = 40 m 2) a = (0−15)/3 = −5 m/s² 3) v² = 0 + 2(2)(100) = 400 → v = 20 m/s

🧠 Chalo Test Karein! (Quick Quiz)

Apne aap ko check karo 👇

Q1. Pehli equation of motion kaunsi hai?

Q2. Kis equation mein TIME nahi hota?

Q3. u = 0, a = 4 m/s², t = 5 s. Distance kitni?

Q4. "Rest se shuru hui" ka matlab?

Q5. 72 km/h kitne m/s hain (numericals ke liye)?

📌 Aage Kya? (Coming Next)

Aakhri lesson mein: Uniform Circular Motion - gol raaste mein chalna, aur ye samajhna ki constant speed par bhi ye "accelerated motion" kyun hai!