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LECTURE NOTES FOR PHYSICS 307: CLASSICAL MECHANICS
TEXT: THORNTON AND MARION'S CLASSICAL MECHANICS OF PARTICLES AND SYSTEMS
SOME USEFUL REFERENCE STUFF: Greek alphabet, metric prefixes, conversion factors
ASIGNMENTS: (Subject to change: check back often.)
HW #3: Due Friday, Sept. 19: Problems 2.24, 27*, 47; 9.55, 62.
*Consider the center of mass of the rope!
HW #4: Due Monday, Sept. 29: Problems 2.7, 9*, 18
*Show that case (b) reduces to case (a) as k goes to zero.
"Dailies":
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Tuesday, Sept. 16: For the Morse potential, V ( r )  V0 e

2 ( r  r0 )
b

 2e

r  r0
b

 , show that the equilibrium is at r=r0

and is stable.
Thursday, Sept. 18: For Physlet #3 at this site, calculate the rate at which mass is ejected from the rocket by
finding the total mass of the rocket plus fuel which keeps the rocket from leaving the launchpad.
Tuesday, Sept. 23: We'll be solving this problem numerically in class Tuesday, but for now please solve it exactly:
y'=dy/dx=2y1/2, with y=1 at x=1.
WEEK 4: GO TO LECTURE 6, 7
LECTURE 6: SECTION 9.11, pp. 58-71
SECTION 9.11: ROCKETS
So why do we spend any time on rocket motion? What's important/useful/interesting about it? Mostly, on the
conceptual level, it's that it is an application of Newton's Second Law in which the mass changes, so that you
have to use the momentum formalism, rather than F=ma. It is also the first example of non-uniform acceleration
that we will deal with using the Force Approach. On the mathematical side, it gives us more experience with
differential equations than we get from simple UAM (uniformly accelerated motion) problems, tools which we will
hone in the next class, on drag.
The first thing to pull out of this material is how to deal with the derivative of p=mv. The simple answer is that we
apply the product rule: d/dt(mv)=m(dv/dt)+(dm/dt)v. In free space, where the net external force is zero, the total
momentum of the rocket plus its exhaust is conserved. We use u to denote the velocity of the exhaust. This is an
important short-term-memory item. Memorize this for the following material, and next to any equations you feel
like circling, mark in what the quantities stand for, especially that u is the fuel velocity. At any rate, we find the
following is true in the case of a rocket firing in free space: (a theoretical ideal which never occurs, except
inasmuch as any external forces are much smaller than the force ("thrust") due to the exhaust escaping)
Rocket in free space:
v  v0  u ln
m0
m
Notice that the original mass, m0 is always larger than the final value, so that the second term on the right hand
side is actually positive. Above, we consider the exhaust plus the rocket as a single "system". We can also think
of the exhaust as exerting an external force on the rocket itself. (Now the "system" is just the rocket.) This force is
called the thrust:
Definition of thrust:  u
dm
dt
We will return to rockets when we get to Chapter 8 (Central forces and orbital dynamics). There we will look at
way more practical questions, like how do you aim a rocket to go from the Earth's orbit to Mars'.
Finally, we will consider the velocity of a rocket as it lifts under the effect of the Earth's gravity:
Vertical ascent in constant gravity:
v   gt  u ln
m0
m
We can also test the motion of a rocket under vertical ascent numerically, as done in this website (Choose #3
from the physlets.), from Evelyn Patterson at the US Air Force Academy. Try an initial mass of fuel of 20000kg,
and see what happens.
LECTURE 7: , PAGES 58-71, these notes, SECTION 3.1 (Return to top.)
DRAG: (Return to pp. 58-71, which we skipped earlier)
I asked the question last time: what was interesting/important/useful about rockets? Now I'll do the same thing
with drag. The reason is that, in both cases, there is all lot of high-powered mathematics thrown at some fairly
esoteric phenomenon. Well, not exactly esoteric, but more complicated than we're used to thinking about. What I
find useful about the discussion of drag in the book is the mathematics. We will have several opportunities to deal
with integrating Newton's laws of motion for cases in which the acceleration is non uniform. This means that we
will have to practice two types of integrations, one in which we integrate F = m d2x/dt2 with respect to time, and
the other in which we use a trick, namely splitting up the dv/dt in F = m dv/dt into a product of dv/dx . dx/dt = dv/dx .
v = v dv/dx. You'll get experience in this section with solving problems in which you use either one integration or
the other. The hard part is to figure out which one to use. If the force has some velocity dependence, then you'll
probably want to use the F = m v dv/dx equation. Otherwise, if the force has either x- or t- dependence but no
velocity dependence, then use F = m d2x/dt2.
Below are the equations that the book derives for specific cases. Jot them down somewhere for future reference,
but the really useful stuff I want you to get out of this section is the ability to integrate F=ma in the two ways I just
mentioned above.
1
ln( 1  kv0 t )
k
gt kv0  g
Linear drag in the presence of gravity: z  h 

1  e kt
k
k2
kV  g
Projectile motion in linear drag medium: T 
1  e kT
gk
Quadratic drag as the only force: x 
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