Momentum (HT only)

What you'll learn

How to calculate momentum using mass and velocity.
Why momentum has direction, so signs matter in collisions.
What conservation of momentum means in a closed system.
How trolley collision data can support the conservation rule.

This part of Forces is Higher Tier only in Combined Science. It can feel a bit more abstract at first, but the method is mostly careful bookkeeping: calculate momenta, keep directions, then compare totals.

Before momentum: mass, velocity and direction

Mass is the amount of matter in an object, measured in kilograms, kg.

Speed tells you how fast something is moving. Velocity tells you speed in a particular direction, measured in metres per second, m/s.

A quantity with size and direction is called a vector. Velocity is a vector. Momentum also turns out to be a vector, so direction matters.

For one-dimensional collision questions, choose one direction as positive. For example:

right is positive, so right-moving velocities are positive
left is negative, so left-moving velocities are negative

This does not mean the object has “less” momentum. A negative sign just shows the direction.

Momentum of a moving object

Momentum is a property of moving objects. A heavy object, or a fast object, has more momentum than a light or slow object.

Definition

Momentum

Momentum is defined by the equation $p = m v$ , where $p$ is momentum in kg m/s, $m$ is mass in kg, and $v$ is velocity in m/s.

The equation is:

p = m v

Momentum is measured in kilogram metre per second, written as kg m/s.

If an object is stationary, its velocity is zero, so its momentum is also zero.

Example

Calculating momentum

A 60 g tennis ball moves to the right at 20 m/s. Calculate its momentum.

Convert the mass into kilograms because the equation needs kg: 60 g = 0.060 kg. The velocity is to the right, so take $v = 20$ m/s.
Substitute into $p = m v$ : $\times 20$ .
Calculate and include the direction: $1.2\ \text{kg m/s}$ to the right.

Tip

Rearranging p = mv

If you need the velocity, use $\frac{p}{m}$ . If you need the mass, use $\frac{p}{v}$ . Always check that mass is in kg before substituting.

Momentum has direction

Because momentum depends on velocity, it has the same direction as the velocity.

This is especially important when objects are moving in opposite directions. You must add momenta using signs, not just add the sizes.

For a system of objects:

momentap_{\text{total}} = p_A + p_B + \text{any other momenta}

If one momentum is negative, it can partly cancel a positive momentum.

Example

Adding momentum in opposite directions

A 3.0 kg trolley moves right at 2.0 m/s. A 2.0 kg trolley moves left at 1.0 m/s. Calculate the total momentum.

Choose right as positive. The first trolley has $v_A = 2.0$ m/s, and the second has $v_B = -1.0$ m/s.
Calculate each momentum: $m/sp_A = 3.0 \times 2.0 = 6.0\ \text{kg m/s}$ and $m/sp_B = 2.0 \times (-1.0) = -2.0\ \text{kg m/s}$ .
Add the momenta: $m/sp_{\text{total}} = 6.0 + (-2.0) = 4.0\ \text{kg m/s}$ . The answer is positive, so the total momentum is 4.0 kg m/s to the right.

Common Mistake

Using speed when direction matters

Speed has no direction. In momentum questions involving collisions, use velocity and include negative signs for motion opposite your chosen positive direction.

Conservation of momentum

To conserve something means the total amount stays the same.

In momentum questions, you usually look at an event, such as a collision between two trolleys.

Definition

Closed system

A closed system is a group of objects where no resultant external force acts during the event. Momentum is not transferred to or from the surroundings.

Key Idea

Conservation of momentum

In a closed system, the total momentum before an event is equal to the total momentum after the event.

During a collision, the objects push on each other. These pushes transfer momentum between the objects, but if the system is closed, the total momentum of all the objects stays the same.

The diagram below shows how to set up a collision using initial velocities and final velocities. Some questions use $u$ for initial velocity and $v$ for final velocity.

Conservation of momentum in a trolley collision

For two objects A and B:

m_A u_A + m_B u_B = m_A v_A + m_B v_B

where $u_A$ and $u_B$ are velocities before the collision, and $v_A$ and $v_B$ are velocities after the collision.

If the objects stick together after the collision, they share one final velocity:

m_A u_A + m_B u_B = (m_A + m_B)v

Example

Trolleys sticking together

A 2.0 kg trolley moves right at 3.0 m/s and collides with a stationary 1.0 kg trolley. They stick together. Calculate their common velocity after the collision.

Choose right as positive and calculate total momentum before: $m/sp_{\text{before}} = 2.0 \times 3.0 + 1.0 \times 0 = 6.0\ \text{kg m/s}$ .
Use conservation of momentum. After the collision, the combined mass is 3.0 kg, so $pafter=3.0vp_{\text{after}} = 3.0v$ .
Set before equal to after and solve: $6.0 = 3.0 v$ , so $v = 2.0$ m/s. The joined trolleys move right at 2.0 m/s.

Common Mistake

Momentum is not kinetic energy

In a closed collision, total momentum is conserved. Kinetic energy does not have to be conserved; some energy may be transferred to thermal stores and sound, especially when objects stick together.

Investigating collisions with trolleys

In the lab, you can investigate conservation of momentum using trolleys on a track.

You can measure velocity using:

light gates: sensors that time how long a card blocks a beam
data loggers: devices that record measurements automatically
ticker timers: devices that mark dots on tape at equal time intervals

A typical method is:

Measure each trolley’s mass using a balance.
Use light gates, data loggers or ticker timers to measure velocities before and after the collision.
Calculate total momentum before the collision.
Calculate total momentum after the collision.
Compare the two totals.

If the system is close to closed, the totals should be very similar.

Example

Using light-gate data

Two 0.50 kg trolleys collide and stick together. Trolley A has a 0.10 m card and passes through a light gate before the collision in 0.20 s. Trolley B is stationary. After the collision, the joined trolleys pass through a light gate in 0.40 s.

Use $\frac{\text{length of card}}{\text{time blocking gate}}$ . Before the collision, trolley A has $\frac{0.10}{0.20} = 0.50$ m/s. After the collision, the joined trolleys have $\frac{0.10}{0.40} = 0.25$ m/s.
Calculate the total momentum before: $m/sp_{\text{before}} = 0.50 \times 0.50 + 0.50 \times 0 = 0.25\ \text{kg m/s}$ .
Calculate the total momentum after: $m/sp_{\text{after}} = (0.50 + 0.50) \times 0.25 = 0.25\ \text{kg m/s}$ . The matching totals support conservation of momentum.

Common Mistake

Real experiments are not perfectly closed

Friction, air resistance and a track that is not level are external effects. In real data, the before and after momenta may be close rather than exactly equal.

Using momentum as a model

A model is a simplified way of representing a situation so you can make predictions.

Momentum is a useful model for collisions because you do not need to describe every tiny detail of the crash. You can compare total momentum before and after the event.

For exam questions, the usual idea is:

after\text{total momentum before} = \text{total momentum after}

Then you substitute the given masses and velocities, keeping signs for direction.

Exam technique

In the exam

Choose a positive direction first, then give every velocity the correct sign.
Calculate total momentum before and total momentum after; do not assume each individual object keeps the same momentum.
Convert units before using equations: mass in kg, velocity in m/s, momentum in kg m/s.

Self review

Check yourself

Why can a small fast object have the same momentum as a large slow object?
If right is positive, what sign should you give to the velocity of an object moving left?
In a closed collision, what quantity is equal before and after the event?

What you'll learn

How to calculate momentum using mass and velocity.
Why momentum has direction, so signs matter in collisions.
What conservation of momentum means in a closed system.
How trolley collision data can support the conservation rule.

Before momentum: mass, velocity and direction

Mass is the amount of matter in an object, measured in kilograms, kg.

Speed tells you how fast something is moving. Velocity tells you speed in a particular direction, measured in metres per second, m/s.

A quantity with size and direction is called a vector. Velocity is a vector. Momentum also turns out to be a vector, so direction matters.

For one-dimensional collision questions, choose one direction as positive. For example:

right is positive, so right-moving velocities are positive
left is negative, so left-moving velocities are negative

This does not mean the object has “less” momentum. A negative sign just shows the direction.

Momentum of a moving object

Momentum is a property of moving objects. A heavy object, or a fast object, has more momentum than a light or slow object.

Definition

Momentum

Momentum is defined by the equation $p = m v$ , where $p$ is momentum in kg m/s, $m$ is mass in kg, and $v$ is velocity in m/s.

The equation is:

p = m v

Momentum is measured in kilogram metre per second, written as kg m/s.

If an object is stationary, its velocity is zero, so its momentum is also zero.

Example

Calculating momentum

A 60 g tennis ball moves to the right at 20 m/s. Calculate its momentum.

Convert the mass into kilograms because the equation needs kg: 60 g = 0.060 kg. The velocity is to the right, so take $v = 20$ m/s.
Substitute into $p = m v$ : $\times 20$ .
Calculate and include the direction: $1.2\ \text{kg m/s}$ to the right.

Tip

Rearranging p = mv

If you need the velocity, use $\frac{p}{m}$ . If you need the mass, use $\frac{p}{v}$ . Always check that mass is in kg before substituting.

Momentum has direction

Because momentum depends on velocity, it has the same direction as the velocity.

This is especially important when objects are moving in opposite directions. You must add momenta using signs, not just add the sizes.

For a system of objects:

momentap_{\text{total}} = p_A + p_B + \text{any other momenta}

If one momentum is negative, it can partly cancel a positive momentum.

Example

Adding momentum in opposite directions

A 3.0 kg trolley moves right at 2.0 m/s. A 2.0 kg trolley moves left at 1.0 m/s. Calculate the total momentum.

Choose right as positive. The first trolley has $v_A = 2.0$ m/s, and the second has $v_B = -1.0$ m/s.
Calculate each momentum: $m/sp_A = 3.0 \times 2.0 = 6.0\ \text{kg m/s}$ and $m/sp_B = 2.0 \times (-1.0) = -2.0\ \text{kg m/s}$ .
Add the momenta: $m/sp_{\text{total}} = 6.0 + (-2.0) = 4.0\ \text{kg m/s}$ . The answer is positive, so the total momentum is 4.0 kg m/s to the right.

Common Mistake

Using speed when direction matters

Speed has no direction. In momentum questions involving collisions, use velocity and include negative signs for motion opposite your chosen positive direction.

Conservation of momentum

To conserve something means the total amount stays the same.

In momentum questions, you usually look at an event, such as a collision between two trolleys.

Definition

Closed system

A closed system is a group of objects where no resultant external force acts during the event. Momentum is not transferred to or from the surroundings.

Key Idea

Conservation of momentum

In a closed system, the total momentum before an event is equal to the total momentum after the event.

During a collision, the objects push on each other. These pushes transfer momentum between the objects, but if the system is closed, the total momentum of all the objects stays the same.

The diagram below shows how to set up a collision using initial velocities and final velocities. Some questions use $u$ for initial velocity and $v$ for final velocity.

Conservation of momentum in a trolley collision

For two objects A and B:

m_A u_A + m_B u_B = m_A v_A + m_B v_B

where $u_A$ and $u_B$ are velocities before the collision, and $v_A$ and $v_B$ are velocities after the collision.

If the objects stick together after the collision, they share one final velocity:

m_A u_A + m_B u_B = (m_A + m_B)v

Example

Trolleys sticking together

A 2.0 kg trolley moves right at 3.0 m/s and collides with a stationary 1.0 kg trolley. They stick together. Calculate their common velocity after the collision.

Choose right as positive and calculate total momentum before: $m/sp_{\text{before}} = 2.0 \times 3.0 + 1.0 \times 0 = 6.0\ \text{kg m/s}$ .
Use conservation of momentum. After the collision, the combined mass is 3.0 kg, so $pafter=3.0vp_{\text{after}} = 3.0v$ .
Set before equal to after and solve: $6.0 = 3.0 v$ , so $v = 2.0$ m/s. The joined trolleys move right at 2.0 m/s.

Common Mistake

Momentum is not kinetic energy

In a closed collision, total momentum is conserved. Kinetic energy does not have to be conserved; some energy may be transferred to thermal stores and sound, especially when objects stick together.

Investigating collisions with trolleys

In the lab, you can investigate conservation of momentum using trolleys on a track.

You can measure velocity using:

light gates: sensors that time how long a card blocks a beam
data loggers: devices that record measurements automatically
ticker timers: devices that mark dots on tape at equal time intervals

A typical method is:

Measure each trolley’s mass using a balance.
Use light gates, data loggers or ticker timers to measure velocities before and after the collision.
Calculate total momentum before the collision.
Calculate total momentum after the collision.
Compare the two totals.

If the system is close to closed, the totals should be very similar.

Example

Using light-gate data

Use $\frac{\text{length of card}}{\text{time blocking gate}}$ . Before the collision, trolley A has $\frac{0.10}{0.20} = 0.50$ m/s. After the collision, the joined trolleys have $\frac{0.10}{0.40} = 0.25$ m/s.
Calculate the total momentum before: $m/sp_{\text{before}} = 0.50 \times 0.50 + 0.50 \times 0 = 0.25\ \text{kg m/s}$ .
Calculate the total momentum after: $m/sp_{\text{after}} = (0.50 + 0.50) \times 0.25 = 0.25\ \text{kg m/s}$ . The matching totals support conservation of momentum.

Common Mistake

Real experiments are not perfectly closed

Friction, air resistance and a track that is not level are external effects. In real data, the before and after momenta may be close rather than exactly equal.

Using momentum as a model

A model is a simplified way of representing a situation so you can make predictions.

Momentum is a useful model for collisions because you do not need to describe every tiny detail of the crash. You can compare total momentum before and after the event.

For exam questions, the usual idea is:

after\text{total momentum before} = \text{total momentum after}

Then you substitute the given masses and velocities, keeping signs for direction.

Exam technique

In the exam

Choose a positive direction first, then give every velocity the correct sign.
Calculate total momentum before and total momentum after; do not assume each individual object keeps the same momentum.
Convert units before using equations: mass in kg, velocity in m/s, momentum in kg m/s.

Self review

Check yourself

Why can a small fast object have the same momentum as a large slow object?
If right is positive, what sign should you give to the velocity of an object moving left?
In a closed collision, what quantity is equal before and after the event?

What you'll learn

Before momentum: mass, velocity and direction

Momentum of a moving object

Momentum has direction

Conservation of momentum

Investigating collisions with trolleys

Using momentum as a model

Review Momentum (HT only) by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Momentum (HT only)

What you'll learn

Before momentum: mass, velocity and direction

Momentum of a moving object

Momentum has direction

Conservation of momentum

Investigating collisions with trolleys

Using momentum as a model

Review Momentum (HT only) by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Momentum (HT only)

What you'll learn

Before momentum: mass, velocity and direction

Momentum of a moving object

Momentum has direction

Conservation of momentum

Investigating collisions with trolleys

Using momentum as a model

Review Momentum (HT only) by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Momentum (HT only) Revision Guide

Momentum (HT only)

What you'll learn

Before momentum: mass, velocity and direction

Momentum of a moving object

Momentum has direction

Conservation of momentum

Investigating collisions with trolleys

Using momentum as a model

Review Momentum (HT only) by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Momentum (HT only) Revision Guide