Momentum

Section 1 of 10Momentum

Momentum

• Momentum the product of an object's mass and its velocity, p = mv.
• The unit is kg m/s, or equivalently N s.

Vector quantity

• Vector quantity momentum has the same direction as the velocity.
• Choose one direction as positive, such as right, and write leftward momentum as negative.

Dependence

• Dependence a heavy slow object can have the same momentum as a light fast object.
• A 1000 kg car at 2 m/s has the same momentum (2000 kg m/s) as a 2 kg object at 1000 m/s.
• Momentum depends on both mass and velocity, so a slow lorry can be harder to stop than a fast cyclist.

Change in momentum

• Δp = mv − mu, the final minus the initial momentum.

Section 2 of 10Force and impulse

Force and impulse

Force as rate of change of momentum

• Force as rate of change of momentum the resultant force equals the change in momentum per unit time, F = Δp ÷ Δt.
• This is Newton's second law in its original form and gives F = ma when the mass is constant.

Impulse

• Impulse the change of momentum, which equals force multiplied by the time for which it acts, impulse = FΔt.
• The unit is N s, the same as kg m/s.

Making the time longer

• Making the time longer the same change in momentum with a longer time means a smaller force.
• A cricketer pulls the hands back when catching a ball, and crumple zones and airbags in cars do the same.

Making the time shorter

• Making the time shorter a shorter contact time gives a larger force, as when a hammer hits a nail.

Calculation route

• Calculation route find Δp from masses and velocities, then divide by the time to get the force.

Section 3 of 10Conservation of momentum

Conservation of momentum

Principle

• Principle in a closed system with no external resultant force, the total momentum before an interaction equals the total momentum after it.

Collision and explosion

• Collision and explosion the principle works for both, although kinetic energy is treated differently.

Method for collision problems

• Method for collision problems follow the steps in order.

1

Choose a positive direction and draw a before and after diagram.

2

Write the masses and velocities, with signs for direction.

3

Calculate the total momentum before.

4

Set the total momentum after equal to this and solve for the unknown.

5

Check the direction of your answer from its sign.

• Equation form
  • m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.
  • For objects that stick together the final velocity is common, so (m₁ + m₂)v on the right-hand side.

Section 4 of 10Types of collision

Types of collision

Elastic collision

• Elastic collision both momentum and kinetic energy are conserved, and the objects bounce apart.

Inelastic collision

• Inelastic collision momentum is conserved but some kinetic energy changes into heat, sound and deformation.

Property	Elastic	Inelastic
Momentum	Conserved	Conserved
Kinetic energy	Conserved	Not conserved
Typical example	Two hard steel balls	Car crash, clay lumps sticking
After collision	Separate	Often join together

Types of collision

Explosions

• Explosions objects start at rest with zero total momentum and fly apart with momenta that cancel.
• A gun recoils backward as the bullet moves forward, and the recoil speed is smaller because the gun has more mass.

Newton's third law link

• Newton's third law link in a collision the forces on the two objects are equal and opposite for the same time,
  • so the changes in momentum are equal and opposite.

Section 5 of 10Linking momentum with kinetic energy and safety

Linking momentum with kinetic energy and safety

Kinetic energy comparison

• KE = ½mv² depends on velocity squared, so doubling speed quadruples energy.

Check for elastic collisions

• Check for elastic collisions calculate the total KE before and after; if equal, the collision is elastic.

Safety applications

• Safety applications car safety features lengthen the time of a collision and so reduce the force on passengers.
• Seat belts stretch slightly, crumple zones deform and airbags cushion the head.

Rockets and jets

• Rockets and jets they work by throwing mass backward at high speed so that the vehicle gains forward momentum.
• The rocket and gas have equal and opposite momentum changes.

Sports

• Sports following through with a bat or racket increases the contact time and imparts more momentum.

Section 6 of 10Worked strategy for collisions in a line

Worked strategy for collisions in a line

Two objects moving towards each other

• Two objects moving towards each other give one a positive velocity and the other a negative velocity.
• A 3 kg trolley at +4 m/s and a 1 kg trolley at −2 m/s have a total momentum of 12 − 2 = +10 kg m/s.

Two objects that bounce apart

• Two objects that bounce apart use momentum conservation for one equation, and the elastic condition, if stated, for the second.
• If kinetic energy is conserved, the relative speed of approach equals the relative speed of separation.

Objects that join

• Objects that join they move together afterwards, so only one unknown velocity remains.

Direction of the answer

• Direction of the answer a positive result means the same direction as the chosen positive direction, a negative result means the opposite direction.

Checking results

• Checking results total momentum after should equal total momentum before, and no object should gain energy from nowhere.
• In an inelastic collision the total kinetic energy after must be less than before.

Energy lost

• Energy lost find KE before minus KE after to see how much changed into thermal and sound energy.
• For the 2.0 kg and 1.0 kg trolleys in the worked example the energy lost is 9.0 − 6.0 = 3.0 J.

Section 7 of 10Momentum in everyday and exam contexts

Momentum in everyday and exam contexts

Stopping distance link

• Stopping distance link a vehicle with large momentum needs a larger impulse to stop, so it needs a longer time or a greater braking force.
• Heavier lorries have longer braking distances than cars moving at the same speed.

Jet and rocket propulsion

• Jet and rocket propulsion hot gas is expelled backward, which gives the rocket forward momentum so that the total remains zero.

Water hose

• Water hose a firefighter feels a backward force because the water leaves with a large momentum each second.

Typical exam wording

• "stationary" means zero initial velocity, "joined together" means a common final velocity,
• and "rebounds" means the velocity reverses direction.

Units check

• Units check a momentum answer should be in kg m/s and an impulse answer in N s, which are the same thing.
• Writing both helps recognise that impulse equals the change in momentum.

Section 8 of 10Worked examples

1. Calculate the momentum of a 1200 kg car travelling at 15 m/s, and of a 60 kg cyclist moving at 8 m/s. [2]2 marks

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2. A 0.50 kg ball moving at 6.0 m/s is hit by a bat and returns at 10 m/s in the opposite direction. The contact lasts 0.020 s. Find the average force. [4]4 marks

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3. A 2.0 kg trolley moving at 3.0 m/s collides with a stationary 1.0 kg trolley and they stick together. Find their common velocity. [3]3 marks

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4. A 0.020 kg bullet is fired at 300 m/s from a 4.0 kg rifle initially at rest. Find the recoil speed. [3]3 marks

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5. Explain why an airbag reduces injury in a crash. [3]3 marks

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Section 9 of 10Key terms

Term	Meaning
Momentum	The product of mass and velocity, p = mv, in kg m/s.
Impulse	The change in momentum, equal to force multiplied by time, in N s.
Conservation of momentum	The total momentum of a closed system is constant.
Elastic collision	A collision in which kinetic energy is conserved.
Inelastic collision	A collision in which kinetic energy is not conserved.
Recoil	The backward motion of a gun or launcher when a projectile is fired.

Section 10 of 10Exam technique

1

Ignoring direction — momentum is a vector, so give opposite directions opposite signs.

2

Saying kinetic energy is always conserved in a collision — only elastic collisions conserve it.

3

Using grams in p = mv — convert mass to kg so the unit is kg m/s.

4

Mixing up impulse and momentum — impulse equals the change in momentum, not the momentum itself.

5

Applying conservation when an external force acts — it holds only for a closed system.