Interacting Systems

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 All kinetic energy lost Kinetic energy unchanged Kinetic energy created Completelyinelastic Perfectly elastic Explosionfrom rest Objects “stick together” Objects “bounce” Objects “explode”

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… L4 L2 L3 L5 L1 Primary Secondary CM