Lesson 7 – Momentum & Collisions

Stage 1: Intermediate Complete Notes
Page 9 – Relative Velocity in Collisions

Relative velocity plays a central role in collision problems. The coefficient of restitution is defined entirely using relative velocities.

                                                 

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1️⃣ What is Relative Velocity?

The velocity of one body as observed from another body is called relative velocity.

Relative velocity of A with respect to B:

v_AB = v_A − v_B

Relative velocity is a vector quantity.

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2️⃣ Relative Velocity Before and After Collision

In collision problems, two relative velocities are important:

• Relative velocity of approach (before collision)
• Relative velocity of separation (after collision)

Relative velocity of approach = u₁ − u₂
Relative velocity of separation = v₂ − v₁

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3️⃣ Coefficient of Restitution Using Relative Velocity

The coefficient of restitution can be written as:

e = (Relative velocity of separation)
    / (Relative velocity of approach)

This definition is valid for:

• Elastic collisions
• Inelastic collisions
• Perfectly inelastic collisions

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4️⃣ Special Case 1: One Body at Rest

If the second body is initially at rest:

u₂ = 0
Relative velocity of approach = u₁

This simplifies calculations in many exam problems.

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5️⃣ Special Case 2: Equal Mass Collision

When two bodies of equal mass collide elastically:

✔ Bodies exchange their velocities
✔ First body comes to rest if second was initially at rest

This result is very useful for quick elimination in MCQs.

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6️⃣ Special Case 3: Perfectly Inelastic Collision

In a perfectly inelastic collision:

✔ Relative velocity of separation = 0
✔ Coefficient of restitution e = 0
✔ Bodies move with common velocity

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7️⃣ Important Exam Tips

✔ Always write relative velocity expression carefully
✔ Watch the sign (+/−) convention
✔ Relative velocity simplifies many collision problems

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📌 Page 9 Summary

✔ Relative velocity = velocity difference
✔ Used directly in coefficient of restitution
✔ Simplifies one-dimensional collision analysis
✔ Essential for IIT & board exams

👉 Next page: Loss of Kinetic Energy in Collisions