Impulse-momentum theorem — derivation

1.4.2 · Physics › Momentum & Collisions

Hum baat kar kya rahe hain?

Hum kyun care karte hain? Kyunki collisions, explosions, kicks, aur catches mein force bahut badi, messy, aur bahut kam time ke liye hoti hai. Hum almost kabhi $\vec{F}(t)$ ko detail mein nahi jaante — lekin hum measure kar sakte hain ki velocity kitni badi. Theorem humein dono ko connect karne deta hai.

Derive kaise karein — Newton ke 2nd law se (first principles se)

Newton ka second law apne original form mein (jis form mein Newton ne actually likha) hai:

${\vec{F}}_{\text{net}}=\frac{d\vec{p}}{dt}$

Yeh form kyun, $F=ma$ kyun nahi? Kyunki $\vec{F}=m\vec{a}$ secretly assume karta hai ki mass constant hai. Momentum form zyada fundamental hai.

Step 1 — Momentum ke rate of change se shuru karo. ${\vec{F}}_{\text{net}}=\frac{d\vec{p}}{dt}$ Yeh step kyun? Yeh force ki definition hai as a momentum-changer; sab kuch yahin se follow hota hai.

Step 2 — Dono sides ko $dt$ se multiply karo (differentials ko alag karo). ${\vec{F}}_{\text{net}}dt=d\vec{p}$ Yeh step kyun? Hum ek time interval mein total effect chahte hain, toh hum chote-chote pushes ko add karne ki taiyari karte hain.

Step 3 — Interval $t_1\to t_2$ par integrate karo. ${\int }_{t_1}^{t_2}{\vec{F}}_{\text{net}}dt={\int }_{{\vec{p}}_1}^{{\vec{p}}_2}d\vec{p}$ Yeh step kyun? Left side har ek choti push ko add karta hai; right side bas momentum mein chote-chote changes ko add karta hai.

Step 4 — Right-hand integral evaluate karo. $\boxed{\vec{J}={\int }_{t_1}^{t_2}{\vec{F}}_{\text{net}}dt={\vec{p}}_2-{\vec{p}}_1=\Delta \vec{p}}$ Yeh step kyun? $d\vec{p}$ ka integral bas final minus initial momentum hota hai.

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Connections

• Newton's Second Law — parent law jisse yeh derive hota hai.
• Conservation of Momentum — kya hota hai jab net external impulse zero ho.
• Collisions — Elastic and Inelastic — contact forces analyze karne ke liye impulse use karta hai.
• Force–Time Graphs — impulse = curve ke neeche ka area.
• Work–Energy Theorem — spatial analogue (force over distance) vs impulse (force over time).
• Center of Mass Motion — external impulse system ke total momentum ko move karta hai.

Concept Map