Elastic Collision Examples in Physics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Elastic Collision.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

A collision in which both the total momentum and the total kinetic energy of the system are fully conserved after impact.

Billiard balls bouncing off each other: the total energy stays the same, nothing is lost to heat or deformation.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Elastic Collision works by defining the interacting system and comparing motion before and after the interaction.

Common stuck point: Students often know a formula related to elastic collision but skip the recognition step: Is the interaction short, collision-like, or rotational, and have I checked whether external forces or torques can be ignored? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Is the interaction short, collision-like, or rotational, and have I checked whether external forces or torques can be ignored?

Worked Examples

Example 1

medium

2\,\text{kg}

ball at

5\,\text{m/s}

hits a

4\,\text{kg}

ball at rest elastically. Find

v_1'

using

v_1' = \frac{m_1 - m_2}{m_1 + m_2}v_1

Answer

v_1' = -\tfrac{5}{3} \text{ m/s} \approx -1.67 \text{ m/s}

First step

Plug in:

v_1' = \frac{2 - 4}{2 + 4}(5) = \frac{-2}{6}(5)

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Example 2

medium

0.2\,\text{kg}

ball at

10\,\text{m/s}

hits a

0.6\,\text{kg}

ball at rest elastically. Find both final velocities.

Example 3

hard

Show: in an elastic 1D collision with

m_2 \gg m_1

and

v_2 = 0

, the light particle bounces back at speed

|v_1|

unchanged.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

In an elastic collision, which two quantities are conserved?

Example 2

easy

A 2 kg ball at

3\,\text{m/s}

collides elastically head-on with an identical 2 kg ball at rest. After an equal-mass elastic collision, find the first ball's velocity.

Example 3

easy

In the same equal-mass elastic collision (2 kg at

3\,\text{m/s}

hits 2 kg at rest), find the second ball's velocity afterward.

Example 4

easy

Total momentum before an elastic collision is

12\,\text{kg·m/s}

. What is it after?

Example 5

easy

Total kinetic energy before an elastic collision is

50\,\text{J}

. What is it after?

Example 6

easy

Is a collision where the objects stick together elastic or inelastic?

Example 7

easy

For a head-on elastic collision, the relative velocity of approach equals the relative velocity of separation. If approach speed is

5\,\text{m/s}

, what is the separation speed?

Example 8

easy

A very light ball bounces elastically off a stationary very heavy wall at

4\,\text{m/s}

. What is its speed after?

Example 9

medium

A 1 kg cart at

4\,\text{m/s}

hits a 3 kg cart at rest elastically (head-on). Find the 1 kg cart's velocity afterward. Use

v_1' = \frac{m_1-m_2}{m_1+m_2}v_1

Example 10

medium

Same collision (1 kg at

4\,\text{m/s}

hits 3 kg at rest, elastic). Find the 3 kg cart's velocity using

v_2' = \frac{2m_1}{m_1+m_2}v_1

Example 11

medium

Verify momentum for the 1 kg/3 kg elastic collision: before

1(4)+3(0)

; after

1(-2)+3(2)

. Are they equal?

Example 12

medium

Verify kinetic energy for the 1 kg/3 kg elastic collision: before

\tfrac12 (1)(4^2)

; after

\tfrac12(1)(2^2)+\tfrac12(3)(2^2)

. Equal?

Example 13

medium

Two equal

2\,\text{kg}

masses approach head-on, one at

5\,\text{m/s}

and the other at

-3\,\text{m/s}

, elastic. Find their velocities afterward.

Example 14

medium

0.2\,\text{kg}

ball moving at

10\,\text{m/s}

bounces elastically straight back off a wall. Find the impulse delivered to the ball.

Example 15

medium

A 4 kg object at

6\,\text{m/s}

collides elastically with a stationary 4 kg object. What fraction of the initial kinetic energy is transferred to the second object?

Example 16

medium

A 3 kg ball at

4\,\text{m/s}

and a 1 kg ball at rest collide elastically. Using the relative-velocity rule, find the separation speed.

Example 17

medium

A 2 kg ball at

5\,\text{m/s}

collides elastically head-on with a 2 kg ball at rest. Find both final velocities (equal masses).

Example 18

challenge

A 1 kg ball at

6\,\text{m/s}

collides elastically head-on with a 2 kg ball at rest. Find both final velocities.

Example 19

challenge

A 2 kg ball at

8\,\text{m/s}

strikes a stationary ball elastically and rebounds at

-4\,\text{m/s}

. Find the mass of the second ball.

Example 20

challenge

A 1 kg ball moving at

v

hits a 1 kg ball at rest elastically; they then both hit a wall. Before any wall contact, the moving ball had

KE = 18\,\text{J}

. Find

v

and confirm the second ball carries

18\,\text{J}

after the first (equal-mass) collision.

Example 21

easy

Define an elastic collision in one sentence.

Example 22

easy

Two billiard balls of equal mass collide elastically head-on. Ball 1 moves at

6\,\text{m/s}

, ball 2 is at rest. What is ball 1's velocity after?

Example 23

easy

0.5\,\text{kg}

ball moves at

4\,\text{m/s}

. What is its kinetic energy?

Example 24

easy

Name a real-world example that is approximately an elastic collision.

Example 25

easy

True or false: in an elastic collision, individual KEs of each object are conserved.

Example 26

easy

Two carts of equal mass approach each other at

3\,\text{m/s}

and

-3\,\text{m/s}

and collide elastically. Find each velocity after.

Example 27

easy

Before an elastic collision,

KE_{total} = 24\,\text{J}

and

p_{total} = 8\,\text{kg·m/s}

. Give both quantities after.

Example 28

medium

Same collision (2 kg at

5\,\text{m/s}

hits 4 kg at rest, elastic). Find

v_2'

using

v_2' = \frac{2 m_1}{m_1 + m_2}v_1

Example 29

medium

Verify momentum for the 2 kg/4 kg elastic collision: compute

p

before and

p

after using

v_1' = -5/3

v_2' = 10/3

Example 30

medium

Verify KE for the 2 kg/4 kg elastic collision (incoming 2 kg at

5\,\text{m/s}

v_1' = -5/3

v_2' = 10/3

Example 31

medium

5\,\text{kg}

block moving at

2\,\text{m/s}

collides elastically head-on with a

5\,\text{kg}

block moving at

-1\,\text{m/s}

. Find both final velocities.

Example 32

medium

A neutron (

1\,\text{u}

) hits a stationary carbon nucleus (

12\,\text{u}

) elastically at

v_0

. Find the neutron's speed after as a fraction of

v_0

Example 33

medium

Ball A moves at

+6\,\text{m/s}

, ball B at

-2\,\text{m/s}

. They collide elastically with equal masses. Find each final velocity.

Example 34

medium

In a 1D elastic collision, what is the fractional KE loss of the system?

Example 35

medium

3\,\text{kg}

glider at

4\,\text{m/s}

hits a

9\,\text{kg}

glider at rest elastically on an air track. Find the

3\,\text{kg}

glider's final velocity.

Example 36

hard

1\,\text{kg}

ball at

6\,\text{m/s}

collides elastically with a

2\,\text{kg}

ball moving at

-3\,\text{m/s}

(head-on). Find the final velocity of the

1\,\text{kg}

ball.

Example 37

hard

Same collision (1 kg at

6\,\text{m/s}

meets 2 kg at

-3\,\text{m/s}

elastic). Find the

2\,\text{kg}

ball's final velocity.

Example 38

hard

0.1\,\text{kg}

ball at

20\,\text{m/s}

hits a stationary block of unknown mass

M

elastically and rebounds at

-10\,\text{m/s}

. Find

M

Example 39

hard

In a 2D elastic collision of equal masses (one initially at rest), the two final velocity vectors are at what angle to each other?

Example 40

hard

0.5\,\text{kg}

ball at

8\,\text{m/s}

elastically strikes a stationary

0.5\,\text{kg}

ball. After collision the first ball moves at

4\,\text{m/s}

at angle

60^\circ

above the line of motion. Find the second ball's speed.

Example 41

hard

A spring-loaded cart of mass

2\,\text{kg}

moving at

3\,\text{m/s}

hits a

1\,\text{kg}

stationary cart. They bounce elastically. Find both final velocities.

Example 42

challenge

A ball of mass

m

at speed

v

collides elastically and head-on with an identical ball at rest, then that ball hits a third identical ball at rest. Find the final speed of the third ball.

Example 43

challenge

Two pucks of mass

m

and

3m

are on a frictionless table. The light puck moves at

v_0

, the heavy puck at

-v_0

. They collide elastically head-on. Find the final velocity of the light puck.

Example 44

challenge

In the previous problem (m and 3m at

\pm v_0

elastic), find the heavy puck's final velocity and verify both

p

and

KE

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Related Concepts

Conservation of Momentum Kinetic Energy Inelastic Collision

Background Knowledge

These ideas may be useful before you work through the harder examples.

conservation of momentumkinetic energy