What Is Momentum?

A moving object has a quantity called momentum.

Momentum describes the motion of an object by combining two important properties:

• its mass
• its velocity

An object with a large mass can have a large momentum.

An object moving at a high velocity can also have a large momentum.

Momentum is represented by the symbol:

p

The relationship between momentum, mass, and velocity is:

p = mv

In equation form:

p = mv

where:

• p = momentum
• m = mass
• v = velocity

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Units of Momentum

Mass is measured in:

kilograms (kg)

Velocity is measured in:

metres per second (m/s)

Therefore:

momentum = kg × m/s

The SI unit of momentum is:

kg·m/s

For example:

p = 30 kg·m/s

Because momentum is a vector, its direction should also be stated when appropriate:

p = 30 kg·m/s east

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Momentum Is a Vector

Momentum is a vector quantity.

This means it has:

• magnitude
• direction

The direction of an object's momentum is always the same as the direction of its velocity.

If a car travels east:

velocity → east

therefore:

momentum → east

If the car reverses direction:

velocity → west

therefore:

momentum → west

This directional property becomes extremely important when analyzing collisions and conservation of momentum.

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Calculating Momentum

The basic momentum equation is:

p = mv

Suppose a 5.0 kg object moves at 4.0 m/s.

p = mv

p = 5.0 × 4.0

p = 20 kg·m/s

If the object moves east:

p = 20 kg·m/s east

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Worked Example 1: A Moving Ball

A 0.50 kg ball travels to the right at 12 m/s.

Calculate its momentum.

p = mv

p = 0.50 × 12

p = 6.0 kg·m/s

Therefore:

Momentum = 6.0 kg·m/s to the right

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Worked Example 2: A Moving Car

A 1200 kg car travels at 20 m/s.

Calculate its momentum.

p = mv

p = 1200 × 20

p = 24,000 kg·m/s

The car has much greater momentum than the ball in the previous example because its mass is much larger.

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What Factors Affect Momentum?

From:

p = mv

we can see that momentum depends on two factors:

1. Mass

2. Velocity

Increasing either one increases the magnitude of momentum.

For example:

A massive truck moving slowly may have more momentum than a small car moving quickly.

A lightweight object moving extremely fast may also have significant momentum.

Both mass and velocity must be considered.

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Effect of Mass on Momentum

Suppose two objects move at the same velocity.

Object A:

m = 2 kg

v = 5 m/s

p = 2 × 5

p = 10 kg·m/s

Object B:

m = 6 kg

v = 5 m/s

p = 6 × 5

p = 30 kg·m/s

The second object has three times the mass and therefore three times the momentum.

If velocity remains constant:

momentum is directly proportional to mass

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Effect of Velocity on Momentum

Now suppose two identical objects move at different velocities.

Object A:

m = 4 kg

v = 3 m/s

p = 12 kg·m/s

Object B:

m = 4 kg

v = 9 m/s

p = 36 kg·m/s

The second object moves three times faster and has three times the momentum.

If mass remains constant:

momentum is directly proportional to velocity

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Doubling Mass or Velocity

Because:

p = mv

if mass doubles while velocity remains constant:

momentum doubles

If velocity doubles while mass remains constant:

momentum doubles

If both mass and velocity double:

momentum becomes four times larger

For example:

Original:

p = mv

Double both:

pnew = (2m)(2v)

pnew = 4mv

Therefore:

pnew = 4p

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Momentum vs Velocity

Momentum and velocity are related, but they are not the same quantity.

Two objects can have the same velocity but different momenta if their masses are different.

Two objects can also have the same momentum but different velocities if their masses are different.

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Same Velocity, Different Momentum

Consider a car and a truck traveling side by side at 20 m/s.

Car:

m = 1000 kg

v = 20 m/s

p = 20,000 kg·m/s

Truck:

m = 8000 kg

v = 20 m/s

p = 160,000 kg·m/s

Both vehicles have the same velocity.

However, the truck has much greater momentum because it has much greater mass.

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Same Momentum, Different Velocities

Suppose two objects both have momentum:

p = 100 kg·m/s

Object A has a mass of 10 kg.

Using:

v = p/m

v = 100 / 10

v = 10 m/s

Object B has a mass of 2 kg.

v = 100 / 2

v = 50 m/s

The lighter object must travel much faster to have the same momentum.

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Rearranging the Momentum Equation

Starting with:

p = mv

To calculate mass:

m = p/v

To calculate velocity:

v = p/m

These forms allow us to solve a variety of momentum problems.

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Worked Example 3: Finding Velocity

An object has:

momentum = 150 kg·m/s

mass = 30 kg

Calculate its velocity.

Use:

v = p/m

v = 150 / 30

v = 5.0 m/s

If the momentum is east:

velocity = 5.0 m/s east

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Worked Example 4: Finding Mass

A moving object has momentum of 240 kg·m/s and velocity of 12 m/s.

Calculate its mass.

Use:

m = p/v

m = 240 / 12

m = 20 kg

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Stationary Objects

A stationary object has:

v = 0

Therefore:

p = mv

p = m(0)

p = 0

So any stationary object has zero momentum, regardless of its mass.

A parked truck may have a huge mass, but if it is not moving:

p = 0

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Direction and Positive/Negative Momentum

Because momentum is a vector, we can use positive and negative signs to represent direction.

Suppose:

Right = positive

Left = negative

A 2 kg object moving right at 5 m/s has:

p = 2(+5)

p = +10 kg·m/s

A 2 kg object moving left at 5 m/s has:

p = 2(−5)

p = −10 kg·m/s

The negative sign does not mean the object has "less" momentum.

It indicates that the momentum points in the negative direction.

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Total Momentum of More Than One Object

When several objects are part of a system, their momenta can be added.

Because momentum is a vector, direction must be included.

Suppose:

Object A:

pA = +20 kg·m/s

Object B:

pB = −12 kg·m/s

Total momentum:

ptotal = pA + pB

ptotal = 20 − 12

ptotal = +8 kg·m/s

The system has a net momentum of:

8 kg·m/s to the right

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Worked Example 5: Objects Moving in Opposite Directions

A 4 kg cart moves right at 6 m/s.

A 3 kg cart moves left at 5 m/s.

Take right as positive.

Cart 1:

p₁ = 4(+6)

p₁ = +24 kg·m/s

Cart 2:

p₂ = 3(−5)

p₂ = −15 kg·m/s

Total:

ptotal = 24 − 15

ptotal = +9 kg·m/s

Therefore:

Total momentum = 9 kg·m/s to the right

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Zero Total Momentum

A system can have moving objects but still have zero total momentum.

Suppose two identical objects move at equal speeds in opposite directions.

Object A:

p = +20 kg·m/s

Object B:

p = −20 kg·m/s

Total:

ptotal = +20 − 20

ptotal = 0

The objects individually have momentum, but the momentum of the entire system is zero.

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Momentum and Newton's Laws

Momentum is closely connected to Newton's Laws.

A force can change an object's momentum.

If a force:

• speeds an object up
• slows an object down
• changes its direction

then its momentum changes.

This idea leads to an important relationship:

Force is related to the rate at which momentum changes.

For constant mass:

F = ma

and since:

p = mv

a change in velocity produces a change in momentum.

This connection becomes especially important when studying impulse.

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Momentum and Stopping Objects

An object with large momentum generally requires a larger force, a longer time, or a longer distance to stop than a similar object with less momentum.

For example, a loaded truck traveling at highway speed has enormous momentum.

Stopping it requires a large change in momentum.

This is one reason heavy vehicles require greater stopping distances than smaller vehicles under comparable conditions.

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Momentum in Sports

Momentum plays an important role in many sports.

Examples include:

• a football player making a tackle
• a soccer player kicking a ball
• a hockey player colliding with another player
• a baseball being struck by a bat
• a tennis ball changing direction after hitting a racket

Mass and velocity determine the momentum of the moving athlete or object.

Changing that momentum requires an interaction involving force.

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Momentum in Vehicle Collisions

Momentum is especially important when analyzing collisions.

Before a collision, each vehicle has momentum determined by its mass and velocity.

During the collision, the vehicles exert forces on each other and their individual momenta change.

However, under suitable conditions, the total momentum of the system remains constant.

This principle is called conservation of momentum and is one of the most important ideas in collision physics.

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Momentum in Space

Momentum is also important in spacecraft motion.

A rocket expels exhaust gases backward at high velocity.

The gases carry momentum backward.

The rocket gains momentum in the opposite direction.

This is closely connected to both:

• Newton's Third Law
• conservation of momentum

Momentum therefore helps explain how rockets can accelerate even in the vacuum of space.

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Momentum and Large Objects

Large objects do not necessarily have large momentum.

Mass alone is not enough.

For example:

A 20,000 kg truck parked at the side of a road:

v = 0

so:

p = 0

A 0.15 kg baseball traveling at 40 m/s:

p = 0.15 × 40

p = 6 kg·m/s

The baseball has momentum while the much more massive stationary truck does not.

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Momentum and Fast Objects

Likewise, high speed alone does not tell us the momentum.

A very small object can travel extremely quickly while still having less momentum than a much larger, slower-moving object.

To compare momentum properly, always consider:

both mass and velocity

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Graphing Momentum and Velocity

For an object of constant mass:

p = mv

Therefore, momentum is directly proportional to velocity.

A graph of momentum against velocity is a straight line passing through the origin.

The gradient represents the object's mass:

gradient = Δp / Δv = m

A steeper graph represents a larger mass.

This gives another way to compare the momentum of objects.

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Did You Know?

A large ship moving relatively slowly can have an enormous momentum because of its huge mass.

This is one reason large ships cannot stop or change direction quickly.

Even at moderate speeds, their enormous mass gives them very large momentum.

Ships must therefore begin slowing or turning well before reaching an obstacle or destination.

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Common Mistakes

Mistake 1: Confusing momentum with velocity

Velocity describes how quickly and in what direction an object moves.

Momentum depends on both mass and velocity.

Mistake 2: Forgetting direction

Momentum is a vector quantity.

Direction must be included when combining momenta.

Mistake 3: Using speed instead of signed velocity

For simple one-dimensional problems, opposite directions should usually be represented using positive and negative velocities.

Mistake 4: Using grams instead of kilograms

Mass should normally be converted to kilograms before calculating momentum.

Mistake 5: Using km/h instead of m/s

SI momentum calculations normally use velocity in m/s.

Mistake 6: Thinking a massive stationary object has momentum

If:

v = 0

then:

p = 0

regardless of mass.

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A Strategy for Solving Momentum Problems

1. Identify the object's mass.
2. Convert mass to kilograms if necessary.
3. Identify its velocity.
4. Convert velocity to m/s if necessary.
5. Choose a positive direction if more than one direction is involved.
6. Use:

p = mv

7. Include the correct unit:

kg·m/s

8. Include direction when required.
9. For several objects, calculate each momentum separately.
10. Add the momenta using their positive and negative signs.

Always check whether the magnitude and direction of the final answer make physical sense.

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Key Terms

Momentum: A vector quantity equal to mass multiplied by velocity.

Mass: A measure of an object's inertia, measured in kilograms.

Velocity: Speed in a particular direction.

Vector: A quantity with both magnitude and direction.

Magnitude: The size of a quantity.

Total momentum: The vector sum of the momenta of all objects in a system.

System: A group of objects considered together when analyzing a physical situation.

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Key Equations

Momentum:

p = mv

Mass:

m = p/v

Velocity:

v = p/m

Total momentum:

ptotal = p₁ + p₂ + p₃ + ...

For one-dimensional motion, use positive and negative signs to represent opposite directions.

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Key Takeaways

• Momentum describes the motion of an object using both mass and velocity.
• Momentum is calculated using p = mv.
• The SI unit of momentum is kg·m/s.
• Momentum is a vector quantity, so it has both magnitude and direction.
• Momentum always points in the same direction as velocity.
• Increasing mass increases momentum when velocity remains constant.
• Increasing velocity increases momentum when mass remains constant.
• A stationary object has zero momentum.
• Two objects can have the same velocity but different momenta.
• Two objects can have the same momentum but different velocities.
• Opposite directions can be represented using positive and negative momentum.
• A system can have zero total momentum even when individual objects are moving.
• Forces change momentum by changing an object's velocity.
• Momentum is important in sports, vehicle safety, collisions, rockets, transportation, and many other physical situations.
• Understanding momentum provides the foundation for studying impulse and conservation of momentum.