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Starting from momentum
Reading 80T: Text to Read
Most physics books tell you about Newton’s law F = m a, and about his third law that forces come in pairs of
equal and opposite ‘action’ and ‘reaction’, and then show how these lead to the conservation of momentum
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This reading summarises how the modern argument goes
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Read it from the top downwards
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Thinking about momentum and forces
Principle 1 symmetry
+v

Principle 2 invariance
–v

+v

–v

seen differently is the same as:

identical objects

+2v
result predictable from symmetry

Conservation of momentum
–p

+p

crunch

Split ‘crunch’ into
forces F on each

mass M

mass m

–p

force –F
acts for
time t

change of momentum = –F t

p
force F =
t

force +F
acts for
time t

same time t
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2
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4
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If two objects come together at the same speed in front of you, and stick together, there is no way the
combined object can go off to the left or right
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But it can change how it looks
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Result: predictions for some simple collisions can be
automatically extended to others
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Changes of
velocity of an object don’t alter when you add or subtract an observer’s velocity from the object’s velocity
before and after the collision
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But this isn’t enough
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Introducing mass
We can define the idea of mass by the ratio of the changes of velocity of two objects in a collision
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m2
v 1
The more massive of the two has the smaller velocity change
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It is the standard kilogram, a lump of metal kept in Paris
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When the masses are unequal, symmetry now says that the changes in the quantity m v must be equal
and opposite
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Give
momentum the symbol p
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Forces act on each
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Just represent
the effect of the other object as a force acting on the object you are concerned with
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Define
the force F which represents the action of the other object as
dp
F 

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dt
Thus we reach Newton’s second law
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Two forces, one on each object, changing
the momentum of both by equal but opposite amounts p, represent one single collision
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Look for the other object
Newton’s law F = m a splits one interaction into two pieces
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Sometimes it’s obvious – a fist for example
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And sometimes it isn’t obvious
at all
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The inference that there
may be a black hole at the centre of our galaxy is another
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This is because interactions are not
regularly ‘taken apart’ or ‘split’
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It turns out that momentum, a bit modified by relativistic ideas, is a central concept in modern
physics, while force is less so
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It relies in the end only on symmetry arguments and on
Galilean invariance: that is, that uniform velocities of observers can’t change events
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But interestingly, it is not in fact quite correct
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It fails to allow for there being a maximum
possible relative velocity, the speed of light
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