Elastic Collision and Coefficient of Restitution

❓ Concept Question

What is the coefficient of restitution and how does it describe collisions?

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🖼 Concept Image

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✍️ Short Concept

This concept is based on:

👉 Collisions
👉 Relative speed
👉 Elastic and inelastic collisions.

Main idea:

$$\boxed{ e= \frac{\text{Relative speed after collision}} {\text{Relative speed before collision}} }$$

Coefficient of restitution tells how much speed is retained after collision.

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🔷 Step 1 — Definition of Coefficient of Restitution 💯

Coefficient of restitution is represented by:

$$e$$

Formula:

$$\boxed{ e= \frac{\text{Relative velocity of separation}} {\text{Relative velocity of approach}} }$$

or simply:

$$\boxed{ e= \frac{\text{After collision}} {\text{Before collision}} }$$

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🔷 Step 2 — Important Range of $e$

Partially Elastic Collision

$$0<e<1$$

Some kinetic energy is lost.

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Perfectly Elastic Collision

$$e=1$$

No kinetic energy loss.

Bodies rebound perfectly.

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Perfectly Inelastic Collision

$$e=0$$

Bodies stick together after collision.

Maximum kinetic energy loss occurs.

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🔷 Step 3 — Ball and Ground Example

When a ball hits ground and bounces back:

If:

$$e=0.8$$

then ball rebounds with reduced speed.

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If:

$$e=0$$

then ball does not bounce.

It sticks or stops after collision.

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🔷 Step 4 — Memory Trick

Easy memory trick:

$$\boxed{ e=\frac{\text{After}}{\text{Before}} }$$

Numerator:

✔ After collision

Denominator:

✔ Before collision

This helps avoid sign confusion in JEE problems.

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🔷 Step 5 — Physical Meaning

Higher value of:

$$e$$

means:

✔ More elastic collision

✔ More rebound speed

✔ Less kinetic energy loss

Lower value means more inelastic nature.

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🔷 Step 6 — JEE Trap Alert 🚨

❌ Relative speed ki jagah individual speed use kar lena

❌ Numerator-denominator ulta kar dena

❌ $e=1$ means momentum conserved only assume kar lena

Remember:

Momentum is conserved in all collisions.

But kinetic energy is conserved only when:

$$\boxed{ e=1 }$$

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✅ Final Takeaway

Coefficient of restitution measures elasticity of collision:

$$\boxed{ e= \frac{\text{Relative speed after collision}} {\text{Relative speed before collision}} }$$

Its value always lies between:

$$0\le e\le1$$

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⭐ Golden JEE Insight

For collision problems:

✔ First apply momentum conservation

✔ Then use:

$$\boxed{ e=\frac{\text{Separation speed}} {\text{Approach speed}} }$$

This combination solves most JEE collision numericals quickly.

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