Momentum

Momentum is a vector

• Velocity carries a direction, and momentum is mass × velocity, so momentum is a vector quantity that points the same way the object is moving
• When working with momentum on a straight line:
  • Pick one direction as positive (rightwards and upwards are the usual conventions)
  • The opposite direction is then negative
  • A momentum that comes out negative simply means the object is moving the opposite way to the direction you chose as positive

When momentum changes

• An object's momentum changes whenever any of the following happen:
  • its velocity increases or decreases (any acceleration along the line of motion)
  • its direction of travel changes (which is also an acceleration, even at constant speed)
  • its mass changes (such as a rocket burning off fuel)

2. Conservation of momentum

The principle

• The principle of conservation of momentum states that, when no outside force acts on a group of interacting objects, the total momentum of the group before the interaction equals the total momentum after the interaction
• "Interaction" in this context can mean:
  • a collision between two objects that come together
  • an explosion where one object breaks into two or more pieces
• Because momentum is a vector, the signs of velocities must be tracked carefully when totals are taken, because opposite-direction velocities can partially or fully cancel

Applying conservation to a collision

• Pick a positive direction for the whole problem at the start
• Write the total momentum before the collision:

p_before = m₁u₁ + m₂u₂

• Write the total momentum after the collision:

p_after = m₁v₁ + m₂v₂

• Set the two equal (the heart of conservation) and solve for the unknown velocity:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

• If two objects start moving towards each other at equal and opposite momenta, the system's total momentum is zero, and the same zero must come out after the collision (e.g. the two objects might end up at rest, or they might fly off back-to-back at matched momenta)

3. Force and rate of change of momentum

Force as the rate of change of momentum

• Combining F = ma and the definition p = mv gives a more general form of Newton's second law:

F = Δp / t = (mv − mu) / t

• where:
  • F = resultant force (N)
  • mv − mu = change in momentum, Δp (kg m/s), i.e. final momentum minus initial momentum
  • t = time over which the change happens (s)
• In words: the resultant force on an object equals its rate of change of momentum

Contact time and impact force

• For a given change in momentum, the force and the contact time are inversely proportional to one another:
  • Halve the contact time and the impact force doubles
  • Triple the contact time and the impact force falls to a third
• Real safety design therefore aims to extend the contact time during a collision to make the impact force as small as possible (see section 5)

Example A — during the same return shot, a 0.40 kg football undergoes the same change in momentum of 8.0 kg m/s in two different scenarios: scenario 1 has a foot-on-ball contact time of 0.050 s, scenario 2 has a contact time of 0.50 s. Compare the average force on the ball.

• Scenario 1: F = Δp / t = 8.0 / 0.050 = 160 N
• Scenario 2: F = Δp / t = 8.0 / 0.50 = 16 N
• The same change in momentum, delivered over ten times longer, gives a force that is ten times smaller

Example B — a 1200 kg car drives at 18 m/s into a wall and rebounds at 4.0 m/s in the opposite direction. The collision lasts 0.20 s. Calculate the average force on the car and state its direction.

• Take the car's forward (towards-wall) direction as positive: u = +18 m/s, v = −4.0 m/s
• Δp = m(v − u) = 1200 × (−4.0 − 18) = 1200 × (−22) = −26 400 kg m/s
• F = Δp / t = −26 400 / 0.20 = −132 000 N
• The minus sign tells you the force on the car points in the opposite direction to its original motion, that is, backwards from the wall, as you would expect

4. Newton's third law

Statement of the law

• Newton's third law of motion states:

Whenever two objects interact, the forces they exert on each other are equal in magnitude and opposite in direction.

• The first two laws are statements about forces on one object at a time; the third law is the one that compares a pair of forces, each on a different object

How to spot a true third-law pair

• A pair of forces is a Newton's-third-law pair if and only if all three of the following are true:
  • The two forces act on two different objects (never on the same object)
  • The two forces are equal in magnitude and opposite in direction
  • The two forces are the same type (e.g. both gravitational, both normal contact, both electrostatic). If you can't name the force type, calling them both "push" or both "pull" is enough

Walking as a third-law example

• When you walk, your foot pushes backwards on the ground (a contact force from foot on ground)
• The ground, by Newton's third law, pushes forwards on your foot with an equal-sized force (a contact force from ground on foot)
• The forwards push on you is what actually accelerates you across the room. The energy to move, though, comes from your own muscles, not from the ground

The book-on-table trap

• A common mistake is to call the weight of a book and the upward reaction of the table a third-law pair. They are not:
  • both arrows act on the same object (the book), failing rule 1
  • they are different types of force (gravity vs contact), failing rule 3
• The book-on-table picture is really a worked illustration of Newton's first law: two balanced forces on one object, so its motion does not change
• The genuine third-law partner of the book's weight is the pull the book exerts back on the Earth, equal in size and opposite in direction, but acting on the Earth rather than on the book

Third law in a collision

• When two objects collide, the force one exerts on the other is matched by an equal and opposite force back:

F_{A on B} = − F_{B on A}

• Equal forces do not mean equal accelerations, because Newton's second law (a = F / m) shows that for a fixed force, a smaller mass picks up a larger acceleration. A small car colliding with a heavy lorry feels the same-size force but is flung backwards far more than the lorry creeps forwards

5. Momentum and safety features

The design idea

• The force a passenger feels in a crash equals their rate of change of momentum, F = Δp / t
• For a given change of momentum (e.g. going from 25 m/s to rest at the same mass), the only thing engineers can change is the time taken to make that change
• Longer contact time → smaller force, and every modern road-safety feature uses this trick

Safety features in vehicles

• Crumple zones: areas at the front and rear of a car designed to deform plastically in a collision rather than rebound. They lengthen the time it takes for the car (and hence the passenger compartment) to come to a halt, lowering the peak force on the occupants. This is why an apparently "small" bump can still leave the bonnet visibly crushed, because the crumple zone is meant to do that
• Seat belts: designed to stretch slightly under the loading of a collision. Stretching keeps the passenger fixed to the seat but lets their body decelerate over a longer time and a few extra centimetres of travel, so the force on the chest is reduced. They also prevent secondary collisions of the passenger with the dashboard or windscreen
• Airbags: gas-filled cushions that inflate within milliseconds of a sensor detecting an impact. The passenger's head and chest strike the airbag rather than the steering wheel or dashboard, and the airbag squashes down over many centimetres, again lengthening the contact time and so reducing the impact force
• Headrests: limit how far the head can snap backwards in a rear-end collision, cushioning the head-and-neck momentum swing and so reducing the chance of whiplash injury

Safety surfaces beyond cars

• Bicycle helmets: the polystyrene foam liner is designed to crush on impact, extending the time over which the head's momentum drops to zero
• Gymnasium and bouldering crash mats: thick, low-density foam pads under climbing walls or below gym apparatus stretch the contact time of a fall from milliseconds to tenths of a second, dropping the peak force on the body to a survivable level

• Playground rubber surfacing: replaces tarmac under climbing frames so a child's fall has a longer stop, reducing the force absorbed by bones and joints

Choosing the right thickness of safety surface

• The contact time needed depends on the change in momentum that has to be absorbed:
  • Large mass and high speed → large change in momentum → thicker / longer-stopping surface needed (a lorry crashing into a barrier needs more deformable material than a bicycle hitting a kerb)
  • Small mass and low speed → smaller change in momentum → thinner surfacing is enough (a child stepping off a low platform onto a soft surface; a thin yoga mat under simple floor exercises)
• Safety features lower the risk of serious injury without erasing it altogether; they swap a brutally large force over a short time for a smaller, but still real, force over a longer one