CLASSICAL MECHANICS & THERMODYNAMICS - Chapter 4, Exercise 3 Solution
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Conceptual: Why Descartes' Undirected "Quantity of Motion" Failed

PROBLEM
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Explain, in your own words, why Descartes' original "quantity of
motion" (mass x speed, with no direction) gave wrong predictions for
collisions between objects moving toward each other, while Huygens'
corrected, directional version (momentum as we use it today) gets
the right answer. Use a head-on collision between two equal-mass
objects moving toward each other at equal speeds as your example.

SOLUTION
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Consider two identical 1 kg balls, each moving at 3 m/s, heading
directly toward each other and colliding head-on.

Descartes' version (mass x speed, no direction):
    Ball A: 1 kg x 3 m/s = 3 (units of "quantity of motion")
    Ball B: 1 kg x 3 m/s = 3 (units of "quantity of motion")
    Total "quantity of motion" before collision = 3 + 3 = 6

Because Descartes' formula ignores direction entirely, it treats both
balls as contributing the SAME positive quantity, even though they are
moving in opposite directions. This gives a total of 6, as if the two
balls were somehow adding their motion together - but two objects
moving toward each other and colliding should intuitively be able to
come to a complete stop (a real, physically observed outcome for two
equal-mass objects meeting head-on at equal speed), which Descartes'
undirected formula cannot correctly predict or explain, since it never
allows the total to reach zero.

Huygens' corrected, directional version (momentum, p = mv, with sign):
    Ball A: 1 kg x (+3 m/s) = +3 kg.m/s   (moving right, say)
    Ball B: 1 kg x (-3 m/s) = -3 kg.m/s   (moving left)
    Total momentum before collision = (+3) + (-3) = 0

With direction correctly included, the two balls' momenta cancel
exactly. Total momentum is zero both before AND after the collision -
correctly allowing (and predicting) the real physical outcome where
two identical balls meeting head-on at equal speed can come to a
complete, simultaneous stop, with zero total momentum before and
after.

ANSWER: Descartes' scalar version cannot represent two objects moving
in opposite directions as anything other than "adding up," so it
cannot correctly predict that they can cancel out and stop. Huygens'
directional version, using positive and negative signs for opposite
directions, correctly allows opposing motions to cancel - matching
what actually happens physically.

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WHY THIS WORKS AS AN ANSWER

This example is deliberately the simplest possible case that exposes
Descartes' real error: a symmetric head-on collision where the
physically correct answer (both objects stopping) requires the total
"quantity of motion" to reach zero - something only possible if
opposite directions are allowed to have opposite signs. Descartes'
mass-times-speed formula, with no sign convention, can only ever add
magnitudes together and so can never predict a total of zero from two
moving objects - exactly the flaw Huygens' vector treatment of
momentum, still in use today, was built to fix.