Physics 3221
Fall Term 2003
Test 2, November 5, 2003
•
This is an open book and notes test lasting 50 minutes.
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There are three problems, divided into subsections. Problem 1 has 3
parts, problem 2 has 4 parts and problem 3 has 1 part. The points for each
part are marked.
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Begin each problem on a fresh sheet of paper. Use only one side of a
sheet of paper.
•
Put your name, the problem number, and the page number on each
sheet.
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To receive partial credit you must explain what you are doing. For the
problems where you have to prove a formula, you have to show your
calculations to receive full credit.
•
Try to draw a box around important results.
There are 3 pages including this page. All the problems are on page 2. Page 3 has
some useful integrals and formulae. Do not forget to look at all parts of the
problems.
Page 1 of 3
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1. A particle of mass m is released from rest at y = 0 and falls vertically (y is the vertical
direction) under the influence of gravity and air resistance (see figure). If the air
resistance is equal to βv2 where v is the velocity of the particle:
a) Draw a free body diagram for the particle and write down the equation of
motion. (2 points)
b) Show that: (3 points)
 βg 
mg
v(t ) =
t 
tanh
m
β


c) Show that the relationship between v and the distance of falling y is given by:
(3 points)
− m   βv 2 

y (v ) =
ln1 −
mg 
2 β  
m
y = 0 and v = 0 at t = 0
2. A single stage rocket has a mass ratio (m0/mf) of 10 (m0 is the total initial mass and mf
is the total final mass after all the fuel has been burnt). The fuel is released with an
exhaust velocity (u) of 4500 m/s. The fuel burns at a constant rate for 300 s (all fuel is
used up in this time of 300 s which is called the burn time tb). The rocket starts from a
launch pad on earth.
a) What is the burn rate (α) in terms of m0? (2 points)
b) Does the rocket take off at t = 0 (t = 0 is the time when m = m0) or does it sit
and burn fuel for a while like your homework problem? (2 points)
c) Assuming constant g, calculate the maximum velocity (vmax) attained by this
single stage rocket. (2 points)
d) If u and m0/mf are kept constant at the values given above and the burn time tb
is reduced does vmax increase or decrease? What is the limit on the maximum
possible velocity as you keep reducing tb? (2 points)
3. Consider a damped harmonic oscillator. In completing four cycles, the amplitude of the
oscillator drops by a factor of e. Starting with the solution for the damped oscillator
problem, find the ratio of the angular frequency of the damped oscillator (ω1) to its
natural angular frequency ωo (ωo is the angular frequency of the same oscillator but
without damping). (4 points)
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