A cricket ball and a lorry travelling at the same speed are very different things to stop. The lorry is far harder to halt — not just because it is heavier, but because it has far more momentum. Momentum connects directly to Newton's second law and to kinetic energy — together they form the core of classical mechanics. Momentum is the product of mass and velocity, and it is one of the most conserved quantities in all of physics. The law of conservation of momentum governs every collision, explosion, and rocket launch — it is why rockets work in the vacuum of space and why snooker balls transfer motion so predictably.
Momentum (p): p = mv, where m is mass (kg) and v is velocity (m/s). A vector quantity in the direction of velocity. Unit: kg·m/s or N·s.
Impulse (J): the change in momentum produced by a force acting over a time interval: J = FΔt = Δp. Unit: N·s. Impulse equals the area under a force-time graph.
Momentum: p = mv
Momentum is a vector — it has the same direction as velocity. A 1,500 kg car at 20 m/s north has momentum 30,000 kg·m/s north. The same car at 20 m/s south has momentum 30,000 kg·m/s south — same magnitude, opposite direction.
Newton's second law is more fundamentally stated in terms of momentum:
The net force equals the rate of change of momentum. The familiar F = ma is a special case when mass is constant: F = m(Δv/Δt) = ma. For variable-mass systems (like a rocket burning fuel), the more general form is required.
Impulse: J = FΔt = Δp
Impulse is the change in momentum produced by a force over a time interval:
The impulse-momentum theorem connects force, time, and momentum change. It reveals a crucial design principle: the same change in momentum can be produced by a large force for a short time, or a small force for a long time.
This is why:
• Car airbags inflate in milliseconds, extending the collision time — the same momentum change (to zero) over a longer time means less average force on the occupant.
• Gymnasts and martial artists bend their knees on landing — extending impact time, reducing peak force.
• Cricket batsmen "give" with the ball on catching — extending contact time, reducing the force on their hands.
• Crash barriers and crumple zones in cars are deliberately deformable — maximising collision time to minimise deceleration force.
The Law of Conservation of Momentum
In an isolated system (no external net force), total momentum is conserved:
where primed quantities are post-collision velocities. This follows directly from Newton's third law: during any collision, the force on object 1 from object 2 is equal and opposite to the force on object 2 from object 1. Their impulses are equal and opposite, so momentum is transferred between them — but the total is unchanged.
Conservation of momentum applies in all directions independently. In 2D collisions, both x and y components are separately conserved.
Types of Collisions
| Type | Momentum conserved? | KE conserved? | Example |
|---|---|---|---|
| Elastic | Yes | Yes | Ideal billiard balls, atomic collisions |
| Inelastic | Yes | No (some lost to heat/sound) | Most real collisions, bouncing ball |
| Perfectly inelastic | Yes | Maximum KE lost | Objects stick together: clay, car crash |
Momentum is always conserved in collisions (no external force). Kinetic energy is only conserved in perfectly elastic collisions. In any real collision, some kinetic energy is converted to heat, sound, and deformation. However, since KE = p²/(2m), knowing post-collision momenta allows you to calculate how much KE was lost — useful in crash analysis and forensics.
Worked Examples
Example 1: Head-on elastic collision
Ball A (2 kg) moving at 6 m/s hits stationary ball B (2 kg). Equal masses in elastic collision: A stops, B moves at 6 m/s.
Example 2: Perfectly inelastic collision
A 1,200 kg car at 15 m/s east collides with a stationary 2,000 kg lorry and they stick together.
Example 3: Rocket propulsion (explosion)
A 500 kg rocket at rest fires exhaust gases: 50 kg of gas ejected at 400 m/s backward. Find the rocket's velocity.
Rocket Propulsion: Momentum Without Anything to Push Against
Rockets work by expelling mass at high velocity backward — by conservation of momentum, the rocket accelerates forward. No air or ground is needed to "push against." This is why rockets work in the vacuum of space. The Tsiolkovsky rocket equation gives the velocity change for a rocket burning fuel:
where v_e is exhaust velocity and m_i/m_f is the initial-to-final mass ratio (the fuel fraction). Reaching orbit requires Δv ≈ 9.4 km/s — which is why most of a rocket's launch mass is fuel.
Frequently Asked Questions
What is momentum in physics?
Momentum (p = mv) is the product of mass (kg) and velocity (m/s). It is a vector quantity in the direction of motion. Unit: kg·m/s. A heavier or faster object has more momentum and is harder to stop. Momentum is conserved in all collisions where no external net force acts.
What is impulse?
Impulse (J = FΔt) is the product of force and the time it acts. It equals the change in momentum: J = Δp = mv_f − mv_i. The same momentum change can be achieved with a large force for a short time or a small force for a long time — the basis of airbags, crumple zones, and catching techniques.
What is conservation of momentum?
In an isolated system (no external net force), total momentum is constant: Σp_before = Σp_after. This applies to all collisions and explosions. It follows from Newton's third law: internal forces between colliding objects are equal and opposite, so their momentum changes cancel.
What is the difference between elastic and inelastic collisions?
Elastic collisions conserve both momentum and kinetic energy (ideal billiard balls, atomic collisions). Inelastic collisions conserve momentum but not KE — some KE converts to heat, sound, or deformation. Perfectly inelastic collisions (objects stick together) conserve momentum and lose the maximum possible KE.
Is momentum a vector or scalar?
Momentum is a vector — it has both magnitude and direction (the same direction as velocity). This is why two identical cars moving at the same speed but in opposite directions have momenta that cancel to zero when added — they have equal magnitudes but opposite directions.
How are momentum and kinetic energy related?
KE = p²/(2m), where p is momentum and m is mass. For a given momentum, a more massive object has less KE. For a given KE, a more massive object has more momentum. In collisions, momentum is always conserved; KE is only conserved in elastic collisions.
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Written by
Dr. James CarterPhysicist and educator with 15+ years teaching classical mechanics and thermodynamics at the university level. Former MIT OpenCourseWare contributor.
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