Physics 201 Lecture 7
Interacting Systems
Impulse changes momentum
• Newton's 2nd law can be written in the following way:
• The mv on the right-hand-side is called momentum
• The left-hand-side is called impulse and characterizes
quick bursts of force – like one sees in a collision
• A rocket engine is graded by its “specific impulse” –
which is related to its effective exhaust velocity
Change in rocket
speed (“delta-v”)
Mass of the
propellant
Exhaust velocity
Mass of the rocket
(without propellant)
Collisions conserve momentum
• So far, we have only dealt explicitly
with single systems – Newton’s 3rd law
is about the forces between systems
– For every action there is an equal and
opposite reaction
– Every force is really one-half of a pair of
forces – an interaction
– This is the modern framework:
interactions exchange momentum
• Forces inside the system redistribute
momentum but the total is conserved
Elastic collisions conserve kinetic energy
• Momentum is insufficient to characterize a collision
– Imagine objects of equal mass that collide with equal but
opposite momentum and stick together
– Consider the opposite scenario: an explosion – the net
momentum of all that shrapnel is also zero
• This spectrum is characterized by a “coefficient of
restitution” which gauges the speed differentials
All kinetic
energy lost
Kinetic energy
unchanged
Kinetic energy
created
Completely
inelastic
Perfectly
elastic
Explosion
from rest
Objects
“stick together”
Objects
“bounce”
Objects
“explode”
Analysis of collisions requires a frame of
reference
• The “center of mass frame” is used when all the velocities
are measured relative to the center of mass of the system
• The advantage of the CM frame is that the net
momentum is zero by definition – the two particles
always have equal and opposite momentum
• Notice that the speed of the center of mass is unchanged
by the collision:
Total momentum is conserved
Total mass is conserved
Lab vs. center of mass frames
• Another common reference is the “lab frame”:
velocities are measured relative to one of the particles
• The velocity of this particle is zero by definition which
make the math quite a bit simpler
Perfectly
elastic
(bounce)
Completely
inelastic
(stick)
2D collisions and scattering experiments
• Conservation laws allow us to
answer many questions without
any knowledge of the interaction or details of the collision
• On the other hand, one can
extract information regarding
the interaction from details of
the collision
• In history, the first example is
Rutherford’s discovery of the
atomic nucleus
2D interactions and bound orbits
• If the kinetic energy of
the particles is less than
their potential energy,
the particles will orbit
• The line between them
always passes through
the center of mass
• Each particle orbits the
center of mass as if the
total mass of the system
were concentrated there
The restricted three-body problem and the
Lagrange points of equilibrium
• The “three-body” problem is literally unsolvable
– The only general option available is to use the “step-forward”
method on a computer (hyperlink to dancing stars)
• But, Lagrange realized that when one mass is very small,
its influence on the other two can be neglected…
L4
L1
L2
Primary
L3
Secondary
L5
CM