1. Physics 201 Lecture 7
Interacting Systems

2. 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
Mass of the rocket (without propellant)
Exhaust velocity

3. 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

4. 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
Completely inelastic
Objects “stick together”
All kinetic energy lost
Perfectly elastic
Objects “bounce”
Kinetic energy unchanged
Explosion from rest
Objects “explode”
Kinetic energy created

5. 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

6. 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)

7. 2D collisions and scattering experiments
Conservation laws allow us to answer many questions without any knowledge of the inter- action 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

8. 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

9. 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…
Primary
Secondary CM L1 L2 L3 L4 L5