Momentum & Collisions

Chapter 2 introduced Newton's own real Second Law formula, F = dp/dt — force as the rate of change of momentum — and then set momentum itself aside to focus on F = ma. This chapter brings momentum back into the centre of the picture, where it earns its own conservation law, every bit as powerful as Chapter 3's conservation of energy.

Momentum: A Vector Quantity

Momentum is defined as mass multiplied by velocity:

Unlike kinetic energy (½mv², which only ever comes out positive), momentum is a vector — it has direction as well as magnitude. A 2 kg ball moving at 3 m/s to the right has momentum +6 kg·m/s; the same ball moving at 3 m/s to the left has momentum −6 kg·m/s. This sign convention is not a mathematical nicety — it is exactly what makes momentum work correctly for the collisions this chapter covers.

A Real Historical Correction: Descartes vs. Huygens

Chapter 2 credited René Descartes with an early forerunner of momentum, his 1644 "quantitas motus" (quantity of motion). Descartes' own version, however, contained a genuine, consequential error: he defined it as simply mass multiplied by speed, with no regard for direction at all — treating a ball moving left and an identical ball moving right at the same speed as having the same "quantity of motion," rather than opposite ones.

This flaw produced real, incorrect predictions for how colliding objects should behave. It was the Dutch physicist Christiaan Huygens who corrected it, through careful study of elastic collisions in the 1650s and 1660s, establishing that momentum genuinely needed direction (a true vector quantity) to correctly predict collision outcomes — the same Huygens whose corrected collision rules Newton would later draw on directly.

Conservation of Momentum

In a closed system — one with no external force acting on it — the total momentum of every object involved stays exactly constant:

Worked Example: A Recoiling Cannon

A 2,000 kg cannon, initially at rest, fires a 10 kg cannonball at 200 m/s. What is the cannon's own recoil velocity?

Total momentum before firing is zero (everything is at rest), so total momentum after firing must also be zero:

The cannon recoils backward at 1 m/s — the negative sign showing it moves opposite to the cannonball, exactly the same physics behind Chapter 2's own rocket-propulsion example, now expressed through conservation of momentum directly.

Elastic vs. Inelastic Collisions

Every collision conserves momentum — but not every collision conserves kinetic energy. That distinction defines two real categories:

• Elastic collision: total kinetic energy is the same before and after. Billiard balls approximate this closely; so do collisions between individual gas molecules.
• Inelastic collision: some kinetic energy converts into heat, sound, or permanent deformation. Momentum is still conserved — only kinetic energy is lost.
• Perfectly inelastic collision: the extreme case, where the colliding objects stick together and move with one shared final velocity — the point of maximum kinetic energy loss consistent with conserving momentum.

Worked Example: A Perfectly Inelastic Collision

A 1,200 kg train car moving at 3 m/s couples with a stationary 800 kg train car. What is their shared velocity after coupling?

Checking kinetic energy: before coupling, KE = ½(1200)(3²) = 5,400 J. After coupling, KE = ½(2000)(1.8²) = 3,240 J. Roughly 2,160 J was lost — converted into the heat, sound, and deformation of the real coupling mechanism — even though momentum (3,600 kg·m/s) was conserved exactly.

Worked Example: An Elastic Collision Between Equal Masses

For a perfectly elastic, head-on collision between two objects of equal mass, the real result simplifies beautifully: the two objects simply exchange velocities. A moving billiard ball striking an identical stationary one transfers essentially all of its velocity to the second ball and comes to a near-complete stop itself — a real, visible demonstration on any pool table.

The Real Story Behind "Newton's Cradle"

The desktop toy of swinging steel balls, ubiquitous on office desks, is popularly named after Isaac Newton — but Newton did not invent it, and the name itself is far younger than the physics it demonstrates. The underlying principle was first demonstrated by French physicist Edme Mariotte, presented to the French Academy of Sciences in 1671 and published in 1673. Newton himself, in the 1687 Principia, credited this real line of work on colliding bodies — alongside Christopher Wren and the same Christiaan Huygens who corrected Descartes above.

The familiar name "Newton's cradle" is far more recent still: it was coined in early 1967 by English actor Simon Prebble for a wooden desktop version made by his own company, Scientific Demonstrations Ltd., first sold through Harrods of London. So the physics is genuinely 17th-century (Mariotte, 1671–1673; acknowledged by Newton, 1687), but the familiar name and desktop toy are a 20th-century invention — nearly three centuries later, and never claimed by Newton himself.

Collision Types Compared

Hands-On Exercises

Next chapter: Circular Motion & Rotational Dynamics — where force and momentum both get rotational counterparts (torque and angular momentum), and centripetal force explains why an orbiting or spinning object doesn't simply fly off in a straight line.