Momentum and Collisions
4. Collisions
Learning outcomes
- I can distinguish between elastic and inelastic collisions.
- I can analyze momentum before and after collisions.
- I can apply conservation of momentum to collisions.
- I can explain energy changes during collisions.
- I can solve one-dimensional collision problems.
What Is a Collision?
A collision is an interaction in which two or more objects exert forces on each other for a relatively short period of time.
Collisions occur in many situations:
- billiard balls striking each other
- carts colliding on a track
- vehicles colliding
- sports balls striking bats or rackets
- atoms and molecules colliding
- subatomic particles interacting
During a collision, momentum can be transferred from one object to another.
If the system is isolated, the total momentum remains constant.
Momentum During a Collision
Momentum is calculated using:
p = mv
where:
- p = momentum in kg·m/s
- m = mass in kg
- v = velocity in m/s
Because velocity is a vector, momentum is also a vector.
Direction therefore matters.
For one-dimensional problems, we normally choose:
right = positive
left = negative
Conservation of Momentum
For an isolated system:
total momentum before = total momentum after
For two objects:
m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f
where:
- m₁ and m₂ are the masses
- v₁ᵢ and v₂ᵢ are the initial velocities
- v₁f and v₂f are the final velocities
This equation is the foundation of collision calculations.
What Is an Isolated Collision?
For momentum conservation to apply directly, the objects should form an isolated system, or a good approximation of one during the collision.
This means the net external impulse during the collision is negligible.
For example, two carts on a low-friction track can often be treated as approximately isolated during their short collision.
The forces the carts exert on each other are internal forces.
Internal Forces During a Collision
Suppose cart A collides with cart B.
Cart A pushes cart B.
At the same time, cart B pushes cart A.
Newton's Third Law tells us that these forces are:
equal in magnitude and opposite in direction.
Therefore, the impulses on the two objects are also equal and opposite.
One object's momentum changes by:
+Δp
while the other's changes by:
−Δp
The momentum is redistributed, but the system's total remains unchanged.
Types of Collision
Collisions are commonly classified according to what happens to kinetic energy.
The two main categories are:
elastic collisions
and:
inelastic collisions
Both can conserve momentum.
The major difference is what happens to the system's kinetic energy.
Elastic Collisions
An elastic collision is a collision in which:
total momentum is conserved
and:
total kinetic energy is conserved
Therefore:
KE before = KE after
where kinetic energy is calculated using:
KE = ½mv²
In an ideal elastic collision, kinetic energy is transferred between objects without a net conversion of the system's kinetic energy into thermal energy, permanent deformation, or other forms.
Examples of Approximately Elastic Collisions
Perfectly elastic collisions are idealizations, but some interactions can approximate them.
Examples include:
- certain particle collisions
- collisions between gas particles in idealized models
- carefully controlled collisions between elastic objects
- some billiard-ball collisions
- certain low-friction laboratory-cart collisions
Real macroscopic collisions usually lose at least a small amount of kinetic energy to other forms.
Example 1: Elastic Collision Between Equal Masses
A 2 kg cart moves right at 4 m/s.
It collides elastically with an identical stationary 2 kg cart.
Before:
Cart A momentum:
pA = 2 × 4 = 8 kg·m/s
Cart B momentum:
pB = 0
Total:
8 kg·m/s
In an ideal head-on elastic collision between equal masses where one object is initially stationary, the moving object can stop while the second object moves away with its velocity.
After:
Cart A:
v = 0 m/s
Cart B:
v = +4 m/s
Final momentum:
(2)(0) + (2)(4) = 8 kg·m/s
Momentum is conserved.
Check the Kinetic Energy
Before:
KE = ½(2)(4²)
KE = 16 J
After:
Cart A:
0 J
Cart B:
KE = ½(2)(4²)
KE = 16 J
Therefore:
KE before = KE after
The collision is elastic.
Inelastic Collisions
An inelastic collision is a collision in which:
momentum is conserved for an isolated system
but:
kinetic energy is not conserved as kinetic energy.
Some kinetic energy is transformed into other forms.
These may include:
- thermal energy
- sound
- vibration
- deformation
- internal energy
Energy Changes During a Collision
Consider two vehicles colliding.
Before the collision, they have kinetic energy because they are moving.
During the collision, some of that kinetic energy may become:
- deformation of the vehicles
- thermal energy
- sound
- vibration
Energy itself is still conserved overall.
It is the kinetic energy that decreases.
This distinction is important:
Momentum conservation
and:
energy conservation
are not the same statement.
Perfectly Inelastic Collisions
A perfectly inelastic collision occurs when the objects:
stick together after the collision.
They therefore share the same final velocity.
For two objects:
m₁v₁ + m₂v₂ = (m₁ + m₂)v
where v is their common final velocity.
A perfectly inelastic collision produces the maximum possible loss of kinetic energy consistent with momentum conservation for the specified initial momentum and masses.
Example 2: Objects Stick Together
A 3 kg cart moves right at 4 m/s.
It collides with a stationary 2 kg cart.
They stick together.
Initial momentum:
pᵢ = (3)(4) + (2)(0)
pᵢ = 12 kg·m/s
Combined mass:
3 + 2 = 5 kg
Conservation of momentum:
12 = 5v
Therefore:
v = 2.4 m/s
The combined carts move:
2.4 m/s to the right
What Happened to the Kinetic Energy?
Before:
KEᵢ = ½(3)(4²)
KEᵢ = 24 J
After:
KEf = ½(5)(2.4²)
KEf = 14.4 J
Change:
24 − 14.4 = 9.6 J
So:
9.6 J
of kinetic energy was transformed into other forms such as deformation, sound, thermal energy, or vibration.
Momentum was still conserved.
Momentum Conserved, Kinetic Energy Reduced
This is one of the most important ideas in collision physics.
For the previous example:
Momentum before:
12 kg·m/s
Momentum after:
12 kg·m/s
But:
Kinetic energy before:
24 J
Kinetic energy after:
14.4 J
Therefore:
momentum conserved
but:
kinetic energy not conserved as kinetic energy
This identifies the collision as inelastic.
Elastic vs Inelastic Collisions
Elastic collision
Momentum:
conserved
Kinetic energy:
conserved
Objects:
usually separate after collision.
Inelastic collision
Momentum:
conserved in an isolated system
Kinetic energy:
decreases as some is transformed into other forms
Objects:
may separate or remain together.
Perfectly inelastic collision
Momentum:
conserved
Kinetic energy:
not conserved as kinetic energy
Objects:
stick together
Inelastic Does Not Always Mean "Stick Together"
This is a common misconception.
Objects can collide, lose some kinetic energy, and then separate.
That collision is still:
inelastic
Only when the objects stick together do we call the collision:
perfectly inelastic.
One-Dimensional Collisions
A one-dimensional collision occurs when all motion is along a single straight line.
Examples include:
- carts moving along a straight track
- two balls colliding head-on
- one cart catching another
- two objects approaching each other directly
One-dimensional problems are easier because we only need one coordinate axis.
Choosing a Direction
Before calculating, choose a positive direction.
For example:
right = positive
Then:
right-moving velocity → positive
left-moving velocity → negative
This sign convention must be used consistently.
Example 3: Two Objects Moving Toward Each Other
A 2 kg cart moves right at 5 m/s.
A 3 kg cart moves left at 2 m/s.
They collide and stick together.
Choose:
right = positive
Initial momentum:
Cart A:
pA = (2)(+5) = +10 kg·m/s
Cart B:
pB = (3)(−2) = −6 kg·m/s
Total:
pᵢ = +4 kg·m/s
Combined mass:
5 kg
Therefore:
5v = 4
v = +0.8 m/s
The positive sign means:
0.8 m/s to the right
Check the Kinetic Energy
Before:
Cart A:
KE = ½(2)(5²) = 25 J
Cart B:
KE = ½(3)(2²) = 6 J
Total:
31 J
After:
KE = ½(5)(0.8²)
KE = 1.6 J
A large amount of kinetic energy has been transformed into other forms.
Because the carts stick together, the collision is:
perfectly inelastic
Example 4: Finding an Unknown Final Velocity
A 4 kg cart moving right at 6 m/s collides with a stationary 2 kg cart.
After the collision, the 4 kg cart moves right at 3 m/s.
Find the final velocity of the 2 kg cart.
Initial momentum:
pᵢ = (4)(6) + (2)(0)
pᵢ = 24 kg·m/s
Final momentum:
p_f = (4)(3) + 2v
Conservation:
24 = 12 + 2v
12 = 2v
v = 6 m/s
Therefore, the second cart moves:
6 m/s to the right
Is Example 4 Elastic?
Momentum conservation alone cannot answer this.
We must compare kinetic energy.
Before:
KEᵢ = ½(4)(6²)
KEᵢ = 72 J
After:
First cart:
KE₁ = ½(4)(3²) = 18 J
Second cart:
KE₂ = ½(2)(6²) = 36 J
Total:
KEf = 54 J
Because:
54 J ≠ 72 J
the collision is:
inelastic
The objects separated, but kinetic energy decreased.
Determining Collision Type from Data
To determine whether a collision is elastic:
Step 1: Check total momentum before.
Step 2: Check total momentum after.
Step 3: Calculate total kinetic energy before.
Step 4: Calculate total kinetic energy after.
If:
KEᵢ = KEf
the collision is elastic.
If:
KEf < KEᵢ
the collision is inelastic.
If the objects also stick together:
perfectly inelastic
Example 5: Identify the Collision
Two 1 kg carts collide.
Before:
Cart A = +3 m/s
Cart B = −1 m/s
After:
Cart A = −1 m/s
Cart B = +3 m/s
Initial momentum:
pᵢ = (1)(3) + (1)(−1)
pᵢ = 2 kg·m/s
Final momentum:
p_f = (1)(−1) + (1)(3)
p_f = 2 kg·m/s
Momentum is conserved.
Now calculate kinetic energy.
Before:
KEᵢ = ½(1)(3²) + ½(1)(1²)
KEᵢ = 4.5 + 0.5
KEᵢ = 5 J
After:
KEf = ½(1)(1²) + ½(1)(3²)
KEf = 0.5 + 4.5
KEf = 5 J
Therefore:
elastic collision
Example 6: Collision with Opposite Final Direction
A 2 kg cart moves right at 8 m/s.
It collides with a 4 kg stationary cart.
After the collision, the 2 kg cart rebounds left at 2 m/s.
Find the velocity of the 4 kg cart.
Take right as positive.
Initial momentum:
pᵢ = (2)(8) = +16 kg·m/s
Final momentum:
p_f = (2)(−2) + 4v
Therefore:
16 = −4 + 4v
20 = 4v
v = +5 m/s
The second cart moves:
5 m/s to the right
Why Rebound Problems Need Care
When an object rebounds, its velocity changes sign.
Before:
+8 m/s
After:
−2 m/s
Failing to include the negative sign would produce an incorrect momentum equation.
Example 7: Finding an Unknown Initial Velocity
A 3 kg cart moving right at an unknown velocity collides with a stationary 2 kg cart.
They stick together and move right at 3 m/s.
Initial:
3u + (2)(0)
Final:
(3 + 2)(3)
Conservation:
3u = 15
u = 5 m/s
The first cart was initially moving:
5 m/s to the right
Example 8: Finding an Unknown Mass
A cart of unknown mass moves right at 4 m/s.
It collides and sticks to a stationary 3 kg cart.
Afterward, they move right at 2 m/s.
Let the unknown mass be m.
Initial momentum:
4m
Final momentum:
(m + 3)(2)
Therefore:
4m = 2m + 6
2m = 6
m = 3 kg
Example 9: Catching Up from Behind
A 2 kg cart travels right at 6 m/s.
A 3 kg cart ahead of it travels right at 2 m/s.
The first cart catches the second and they stick together.
Initial momentum:
pᵢ = (2)(6) + (3)(2)
pᵢ = 12 + 6
pᵢ = 18 kg·m/s
Combined mass:
5 kg
Therefore:
5v = 18
v = 3.6 m/s
Both carts move right at:
3.6 m/s
Notice that the final speed lies between the two initial speeds.
A Useful Reasonableness Check
For two objects moving in the same direction that stick together, the final velocity should normally lie between their initial velocities.
For example:
6 m/s and 2 m/s
produced:
3.6 m/s.
If your answer were:
12 m/s
you should immediately suspect an error.
Physics answers should always be checked for reasonableness.
Example 10: Equal and Opposite Momentum
A 5 kg cart moves right at 4 m/s.
Another cart moves left with momentum:
−20 kg·m/s
First cart:
p = 5 × 4 = +20 kg·m/s
Total initial momentum:
+20 + (−20) = 0
If the carts stick together:
total final momentum = 0
Therefore:
final velocity = 0 m/s
The combined object is stationary after the collision.
Momentum Can Be Zero While Kinetic Energy Is Not
Before the previous collision, both objects were moving.
Therefore, both had kinetic energy.
Yet their total momentum was:
zero
This is possible because momentum has direction.
Kinetic energy does not have direction.
This demonstrates an important difference between momentum and kinetic energy.
Momentum Is a Vector, Energy Is a Scalar
Momentum:
vector
Direction matters.
Kinetic energy:
scalar
Direction does not matter.
For example:
A 2 kg object moving at +5 m/s and a 2 kg object moving at −5 m/s have opposite momenta:
+10 kg·m/s
and:
−10 kg·m/s
But both have the same kinetic energy:
25 J
Where Does the "Lost" Kinetic Energy Go?
Kinetic energy is not destroyed.
During an inelastic collision it is transformed.
Possible forms include:
Thermal energy
Materials heat slightly.
Sound
Vibrations produce sound waves.
Deformation
Objects bend, dent, compress, or break.
Internal energy
Microscopic motion and structural changes increase.
Total energy is conserved even though kinetic energy decreases.
Deformation and Collision Energy
Consider a car crash.
Some of the vehicle's kinetic energy is transformed as parts of the vehicle deform.
This is one reason vehicles use structures designed to deform in controlled ways during collisions.
The collision is highly inelastic.
Momentum conservation can still be used for an appropriately chosen system over the short collision interval, even though large amounts of kinetic energy are transformed.
Billiard Balls
Billiard-ball collisions can approximate elastic behaviour under suitable conditions.
When one ball strikes another:
- momentum is transferred
- the balls usually separate
- much of the kinetic energy remains kinetic
However, real collisions still produce some sound, heat, and deformation, so they are not perfectly elastic.
Newton's Cradle
A Newton's cradle demonstrates how momentum and kinetic energy can be transferred through collisions.
The collisions approximate elastic behaviour.
Momentum and kinetic energy pass through the interacting balls, causing a ball on the opposite side to move.
Real Newton's cradles gradually stop because some energy is lost from the mechanical motion through sound, air resistance, deformation, and friction.
Collision Forces and Impulse
During a collision, forces can be very large because momentum changes over a short time.
Impulse is:
J = Δp
and:
J = FΔt
for a constant or average force description.
Therefore:
FΔt = Δp
This connects collision physics with impulse.
A large momentum change over a very short time can produce a large average force.
Collision Time and Safety
Suppose the same momentum change occurs over two different time intervals.
If the collision time increases:
average force decreases
for the same momentum change.
This principle helps explain the role of:
- airbags
- seat belts
- crumple zones
- helmets
- protective padding
- crash mats
These systems can increase the time over which momentum changes, reducing average force.
Collisions in Sports
Collision physics appears throughout sport.
Examples include:
- bat and baseball
- racket and tennis ball
- golf club and golf ball
- foot and football
- hockey stick and puck
During these interactions:
- momentum changes
- momentum is transferred
- forces act over short time intervals
- energy may be stored temporarily through deformation
Understanding collisions helps explain both performance and protective equipment.
Collisions at the Particle Scale
Collision analysis is also fundamental in atomic, nuclear, and particle physics.
Scientists use conservation laws to analyze interactions between particles.
Momentum before and after an interaction provides evidence about what occurred.
Sometimes conservation laws can even indicate that an unseen particle carried away momentum.
A General One-Dimensional Collision Strategy
Use the following process for collision problems.
Step 1: Identify the objects.
Determine which objects belong to the system.
Step 2: Decide whether momentum conservation applies.
Check whether external impulse is negligible.
Step 3: Choose a positive direction.
Usually right = positive.
Step 4: Record the initial velocities.
Include negative signs when necessary.
Step 5: Calculate initial momentum.
Use:
p = mv
Step 6: Write the momentum conservation equation.
Σpᵢ = Σpf
Step 7: Use information about the collision.
If objects stick:
they share one final velocity.
If elastic:
kinetic energy is also conserved.
Step 8: Solve for the unknown.
Step 9: Interpret the sign.
Positive and negative indicate direction.
Step 10: Check momentum.
Confirm total momentum before equals total momentum after.
Step 11: If required, calculate kinetic energy.
Use:
KE = ½mv²
to classify or analyze the collision.
Worked Problem 1
A 6 kg cart moving right at 5 m/s collides with a stationary 4 kg cart.
They stick together.
Initial momentum:
pᵢ = (6)(5)
pᵢ = 30 kg·m/s
Combined mass:
10 kg
Therefore:
30 = 10v
v = 3 m/s
Final velocity:
3 m/s right
Because they stick together:
perfectly inelastic collision
Worked Problem 2
A 2 kg cart moves right at 7 m/s.
A 3 kg cart moves left at 2 m/s.
They stick together.
Initial momentum:
pᵢ = (2)(7) + (3)(−2)
pᵢ = 14 − 6
pᵢ = 8 kg·m/s
Combined mass:
5 kg
Therefore:
5v = 8
v = 1.6 m/s
The positive answer means:
1.6 m/s right
Worked Problem 3
A 5 kg cart moving right at 4 m/s strikes a stationary 5 kg cart.
Afterward, the first cart stops.
Find the second cart's velocity.
Initial momentum:
20 kg·m/s
After:
(5)(0) + 5v = 20
Therefore:
v = 4 m/s
If kinetic energy is also unchanged, this is consistent with an ideal elastic collision between equal masses.
Worked Problem 4
A 3 kg cart moves right at 6 m/s and collides with a 2 kg stationary cart.
Afterward, the 3 kg cart moves right at 2 m/s.
Find the second cart's velocity.
Initial momentum:
18 kg·m/s
Final:
(3)(2) + 2v
Therefore:
18 = 6 + 2v
12 = 2v
v = 6 m/s
The second cart moves:
6 m/s right
Worked Problem 5: Classify the Collision
Using Worked Problem 4:
Initial kinetic energy:
KEᵢ = ½(3)(6²)
KEᵢ = 54 J
Final kinetic energy:
First cart:
½(3)(2²) = 6 J
Second cart:
½(2)(6²) = 36 J
Total:
42 J
Because:
54 J ≠ 42 J
kinetic energy decreased.
Therefore:
inelastic collision
The carts did not stick, so it is inelastic but not perfectly inelastic.
Worked Problem 6: Rebound
A 1 kg ball moves right at 10 m/s and collides with a stationary 4 kg object.
Afterward, the ball rebounds left at 2 m/s.
Find the 4 kg object's velocity.
Initial momentum:
+10 kg·m/s
Final:
(1)(−2) + 4v
Therefore:
10 = −2 + 4v
12 = 4v
v = 3 m/s
The 4 kg object moves:
3 m/s right
Worked Problem 7: Unknown Initial Speed
A 4 kg cart moving right collides with a stationary 6 kg cart.
They stick together and move right at 2 m/s.
Initial velocity of the 4 kg cart = u.
Initial momentum:
4u
Final momentum:
(4 + 6)(2) = 20
Therefore:
4u = 20
u = 5 m/s
Worked Problem 8: Collision Type from Energy Data
A collision has:
Initial kinetic energy:
120 J
Final kinetic energy:
120 J
and total momentum is conserved.
Classification:
elastic
If instead the final kinetic energy were:
85 J
the collision would be:
inelastic
The missing 35 J of kinetic energy would have been transformed into other forms.
Collision Diagrams
Drawing a simple before-and-after diagram can prevent many mistakes.
Include:
Before
- mass of each object
- velocity magnitude
- direction
After
- mass of each object
- final velocity
- direction
Then choose a sign convention before writing equations.
Common Mistakes
Mistake 1: Saying only elastic collisions conserve momentum
Momentum is conserved in both elastic and inelastic collisions when the system is isolated.
Mistake 2: Saying energy is lost
Total energy is conserved.
In an inelastic collision, some kinetic energy is transformed into other forms.
Mistake 3: Assuming every inelastic collision involves sticking
Only a perfectly inelastic collision requires the objects to stick together.
Mistake 4: Ignoring direction
Left-moving velocities usually need negative signs when right is chosen as positive.
Mistake 5: Adding speeds instead of momenta
Calculate:
mv
for each object.
Mass matters.
Mistake 6: Forgetting to combine masses
When objects stick:
final mass = m₁ + m₂
Mistake 7: Using kinetic energy conservation for every collision
Only use:
KEᵢ = KEf
when the collision is elastic or when the data establish that kinetic energy is conserved.
Mistake 8: Using momentum conservation when a large external impulse acts on the chosen system
First define the system and decide whether it is isolated or approximately isolated during the collision.
Mistake 9: Forgetting that velocity can change sign
A rebound changes direction.
For example:
+6 m/s → −2 m/s
is a significant velocity and momentum change.
Did You Know?
Collision analysis has played an important role in the development of physics.
Modern particle accelerators create controlled collisions between particles moving at extremely high speeds.
Scientists analyze the momentum and energy of the particles produced after these collisions.
Conservation laws allow researchers to reconstruct what happened during interactions that occur on scales far too small to observe directly.
The mathematics becomes more advanced at very high speeds because relativistic momentum and energy must be used, but the conservation principles remain fundamental.
Key Terms
Collision: A short interaction during which objects exert forces on one another.
Momentum: A vector quantity equal to mass multiplied by velocity.
Elastic collision: A collision in which both total momentum and total kinetic energy are conserved.
Inelastic collision: A collision in which momentum is conserved in an isolated system but kinetic energy is transformed into other forms.
Perfectly inelastic collision: An inelastic collision in which the objects stick together.
Isolated system: A system experiencing negligible net external impulse during the interaction.
Kinetic energy: Energy associated with motion.
Impulse: Change in momentum produced by a force acting over time.
Rebound: Motion away from a collision in the opposite direction to the object's initial motion.
Deformation: A change in an object's shape.
One-dimensional collision: A collision in which motion occurs along a single straight line.
Key Equations
Momentum:
p = mv
Conservation of momentum:
Σp(before) = Σp(after)
Two-object collision:
m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f
Objects that stick together:
m₁v₁ + m₂v₂ = (m₁ + m₂)v
Kinetic energy:
KE = ½mv²
Elastic collision:
total KE before = total KE after
Impulse:
J = Δp
Average-force form:
FΔt = Δp
Key Takeaways
- A collision is a short interaction in which objects exert forces on one another.
- Momentum is a vector, so direction must be included.
- For an isolated system, total momentum before a collision equals total momentum after.
- Momentum is conserved in both elastic and inelastic collisions when the system is isolated.
- In an elastic collision, total kinetic energy is also conserved.
- In an inelastic collision, some kinetic energy is transformed into thermal energy, sound, deformation, vibration, or other forms.
- Total energy is still conserved.
- In a perfectly inelastic collision, the objects stick together and share a final velocity.
- Not all inelastic collisions involve objects sticking together.
- Momentum conservation and kinetic-energy conservation are different principles.
- A collision can have zero total momentum while the objects still have kinetic energy.
- Opposite directions must be represented using opposite signs.
- Rebounding objects change the sign of their velocity.
- Collision problems can involve unknown final velocity, initial velocity, momentum, or mass.
- Checking kinetic energy before and after can determine whether a collision is elastic or inelastic.
- Momentum transfer during collisions can be explained using Newton's Third Law and impulse.
- Increasing collision time can reduce average force for a given momentum change, which is important in safety systems.
- Collisions are important in transport, sport, engineering, atomic physics, and particle physics.
- A reliable one-dimensional collision method is:
define the system → choose a positive direction → calculate initial momentum → apply momentum conservation → solve the unknown → check direction → compare kinetic energy if needed → classify the collision.