In simple terms
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Linear momentum and its conservation
Cambridge 9702 Paper 2 — Linear momentum and its conservation (3.3). Senpai Corner diagram-backed pilot with premium structure and live visuals.
- 1
Momentum (p) is the product of mass (m) and velocity (v).
- 2
The unit of momentum is the kilogram-metre per second (kg m/s).
- 3
As a vector, momentum's direction is the same as the velocity's direction.
- 4
It is a measure of an object's 'quantity of motion'.
What this topic covers
The official Cambridge syllabus points this lesson works through.
- 3.3.1
State the principle of conservation of momentum
- 3.3.2
Apply the principle of conservation of momentum to solve simple problems, including elastic and inelastic interactions between objects in both one and two dimensions (knowledge of the concept of coefficient of restitution is not required)
- 3.3.3
Recall that, for an elastic collision, total kinetic energy is conserved and the relative speed of approach is equal to the relative speed of separation
- 3.3.4
Understand that, while momentum of a system is always conserved in interactions between objects, some change in kinetic energy may take place
Explore the concept
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Key formulas
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Tap a symbol — great for exam definitions
Tap a symbol — great for exam definitions
Tap a symbol — great for exam definitions
Full topic notes
Formal explanation with the rigour you need for the exam.
What is Linear Momentum?
Linear momentum, denoted by the symbol , is a measure of an object's 'quantity of motion'. It’s a vector quantity, which means it has both a magnitude and a specific direction. The direction of an object's momentum is always the same as the direction of its velocity. Think of a heavy lorry moving slowly versus a small car moving quickly – which has more 'oomph' or momentum?
Momentum (p) is the product of mass (m) and velocity (v).
The unit of momentum is the kilogram-metre per second (kg m/s).
As a vector, momentum's direction is the same as the velocity's direction.
It is a measure of an object's 'quantity of motion'.
Momentum, Force, and Impulse
Newton's Second Law can be expressed more fundamentally in terms of momentum. Force is the rate at which an object's linear momentum changes over time. This leads to the concept of impulse, which is the change in momentum itself. Impulse is also the product of the average force and the time over which it acts, a crucial idea for understanding impacts.
Impulse (J) is the change in momentum (Δp).
It is calculated as the product of average force and time duration (FΔt).
The unit of impulse is the Newton-second (Ns), which is equivalent to kg m/s.
The area under a force-time graph represents the impulse delivered.
Remember that (delta) signifies 'change in'. So, means final momentum minus initial momentum (). This formula highlights that a larger force causes a faster change in momentum, and a force applied for a longer time causes a larger change in momentum.
The Principle of Conservation of Momentum
This principle is one of the most fundamental laws in physics. It states that for a closed system, where no net external forces act, the total linear momentum of the system remains constant. This means the total momentum before an event, such as a collision or an explosion, will be exactly equal to the total momentum after the event. The individual momenta of objects within the system might change, but their vector sum stays the same.
Total momentum of a closed system remains constant.
Applies when no net external forces act on the system.
Total momentum before = Total momentum after (as a vector sum).
Crucial for analysing collisions and explosions.
Elastic vs. Inelastic Collisions
While linear momentum is always conserved in a closed system, the total kinetic energy of the system might not be. This difference allows us to classify collisions into two main types: elastic and inelastic. It's crucial for understanding the energy transformations that occur during impacts.
Elastic Collision: Total kinetic energy (KE) of the system IS conserved.
Elastic Collision: Relative speed of approach equals relative speed of separation.
Inelastic Collision: Total kinetic energy (KE) of the system is NOT conserved.
Perfectly Inelastic Collision: A specific type where objects stick together after impact, resulting in the maximum possible loss of KE.
Momentum: Is always conserved in BOTH types of collisions (in a closed system).
A common misconception is that kinetic energy is always conserved if momentum is. Remember, only in perfectly elastic collisions is kinetic energy conserved. In all inelastic collisions, momentum is conserved, but kinetic energy is not.
Explosions and Recoil
The principle of conservation of momentum is also perfectly demonstrated in explosions or recoil scenarios. Here, an object initially at rest breaks apart into two or more pieces. Since the initial momentum of the system is zero, the vector sum of the momenta of all the pieces after the explosion must also be zero. This means the pieces will fly off in opposite directions (in a two-body explosion) to keep the total momentum conserved.
Initial Momentum = 0 Final Momentum =
For a two-body explosion from rest, , which means . The negative sign indicates that the velocities are in opposite directions. This is the principle behind rocket propulsion and the recoil of a gun.
Worked examples
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A trolley of mass 2.0 kg moving at 3.0 m/s collides head-on with a stationary trolley of mass 4.0 kg. If the two trolleys stick together after the collision, what is their common final velocity?
- 1
Identify initial conditions:
A ball of mass 0.50 kg moving at 4.0 m/s collides head-on with a stationary ball of mass 0.30 kg. The collision is perfectly elastic. Calculate the final velocities of both balls.
- 1
Define variables & initial conditions:
How it all connects
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Glossary
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Quick check
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Revision flashcards
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What is the formula for calculating linear momentum?
Linear momentum () is calculated as the product of mass () and velocity (), so .
Key takeaways
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- ✓
Momentum (p) is the product of mass (m) and velocity (v).
- ✓
The unit of momentum is the kilogram-metre per second (kg m/s).
- ✓
As a vector, momentum's direction is the same as the velocity's direction.
- ✓
It is a measure of an object's 'quantity of motion'.
Practice — then mark it
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