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Topic 10 of 20

Collisions

Collisions are the canonical application of momentum conservation. The type of collision — elastic or inelastic — determines whether kinetic energy is also conserved, leading to different sets of equations. Understanding collisions is essential in everything from billiards to particle physics.

Key Concepts

Elastic Collision

Both momentum AND kinetic energy are conserved. Examples: billiard balls, atomic/subatomic collisions. Provides two equations for two unknowns (the final velocities).

Inelastic Collision

Momentum is conserved but kinetic energy is NOT (some is lost to heat, sound, or deformation). Most everyday collisions are inelastic.

Perfectly Inelastic Collision

Objects stick together after collision — maximum kinetic energy loss consistent with momentum conservation. Final velocity:

v_f = (m_1v_{1i} + m_2v_{2i})/(m_1+m_2)

Coefficient of Restitution

Ratio of relative speed after to before collision:

e = |v_{1f}-v_{2f}|/|v_{1i}-v_{2i}|

. Elastic:

e=1

; perfectly inelastic:

e=0

; most collisions:

0 < e < 1

2D Collisions

Momentum conservation applies independently in both the x and y directions. Kinetic energy conservation (for elastic) gives a third equation. Use components throughout.

Center-of-Mass Frame

In the CM frame, total momentum is zero. Elastic collisions in the CM frame simply reverse the velocities of each particle. Useful for analyzing complex collisions.

Key Equations

Perfectly inelastic final velocity

v_f = \frac{m_1 v_{1i} + m_2 v_{2i}}{m_1 + m_2}

Objects stick together; derived directly from momentum conservation.

Elastic collision: final velocities (1D)

v_{1f} = \frac{m_1-m_2}{m_1+m_2}v_{1i} + \frac{2m_2}{m_1+m_2}v_{2i}

One of two symmetric equations for 1D elastic collisions (swap 1↔2 for v2f).

Kinetic energy (elastic)

K_i = K_f \quad\Longrightarrow\quad \tfrac{1}{2}m_1v_{1i}^2 + \tfrac{1}{2}m_2v_{2i}^2 = \tfrac{1}{2}m_1v_{1f}^2 + \tfrac{1}{2}m_2v_{2f}^2

Additional constraint for elastic collisions; used together with momentum conservation.

KE lost (inelastic)

\Delta K = K_f - K_i < 0

Energy lost to internal degrees of freedom; maximum loss in a perfectly inelastic collision.

Worked Example

Perfectly Inelastic Collision

Problem

A 1200 kg car moving at 15 m/s east collides with a 1800 kg truck at rest. They stick together. Find the final velocity.

Solution

Take east as positive. Momentum conservation:

m_1 v_{1i} + m_2 v_{2i} = (m_1+m_2)v_f

1200\times15 + 1800\times 0 = (1200+1800)\,v_f

v_f = \frac{18000}{3000} = 6.0 \text{ m/s (east)}

KE lost: $K_i = \frac{1}{2}(1200)(15)^2 = 135{,}000$ J; $K_f = \frac{1}{2}(3000)(6)^2 = 54{,}000$ J. Lost: 81 kJ.

Answer Final velocity = 6.0 m/s east. 81 kJ of kinetic energy is lost.

Practice

Exercises

7 problems

1 of 7

A $2 \text{ kg}$ ball at $6 \text{ m/s}$ collides and sticks with a $4 \text{ kg}$ ball at rest. What is the final speed?

Your answer

m/s

2 of 7

Same collision ( $2 \text{ kg}$ at $6 \text{ m/s}$ sticks to $4 \text{ kg}$ at rest). How much kinetic energy is lost?

Your answer

3 of 7

A $1 \text{ kg}$ ball at $8 \text{ m/s}$ has an elastic head-on collision with an identical $1 \text{ kg}$ ball at rest. What is the speed of the first ball after the collision?

Your answer

m/s

4 of 7

A $1 \text{ kg}$ ball at $8 \text{ m/s}$ has an elastic head-on collision with a stationary $3 \text{ kg}$ ball. What is the speed of the $1 \text{ kg}$ ball after the collision?

Your answer

m/s

5 of 7

A $4 \text{ kg}$ cart at $5 \text{ m/s}$ has a perfectly inelastic collision with a $6 \text{ kg}$ cart moving at $2 \text{ m/s}$ in the same direction. What is their final speed?

Your answer

m/s

6 of 7

Two equal masses collide elastically head-on. Ball 1 moves at $4 \text{ m/s}$ and ball 2 at $-2 \text{ m/s}$ (opposite direction). What is ball 1's velocity after the collision?

Your answer

m/s

7 of 7

The coefficient of restitution between two balls is $0.6$ . If their relative approach speed is $10 \text{ m/s}$ , what is their relative separation speed after the collision?

Your answer

m/s

Key Takeaways

Momentum is always conserved in collisions (no net external force). Kinetic energy may or may not be conserved.
Elastic: both KE and momentum conserved. Inelastic: only momentum conserved. Perfectly inelastic: objects stick together.
For 1D elastic collision with $m_2$ at rest: equal masses → exchange velocities; heavy hitting light → projectile continues; light hitting heavy → projectile bounces back.
In 2D, apply momentum conservation separately in x and y. Count equations vs. unknowns before solving.
The kinetic energy lost in a perfectly inelastic collision is the maximum possible loss for given initial conditions.