Math 2250 Lab 2
Name/Unid:
Due Date: 1/23/2014
Class ID:
Section:
1. (20 points) Electric Circuits
For the RC-circuit shown below, suppose constant input voltage V (t) = V0 . According to Kirchhoff’s current law, VR (t) + VC (t) = V (t), where VR (t) = I(t)R = RCVC0 (t)
is the voltage across the resistor and VC (t) is the voltage across the capacitor.
−t
(a) Show that VC (t) = −V0 e RC + V0 given VC (0) = 0.
(b) Assume V0 = 200 volts. Find VR (t). Provide a technology answer check.
Background: Let V (t) represent input voltage in volts. Then V (t) ≡ V0 represents
a constant applied voltage, C represents capacitance in farads, R represents resistance
in ohms, and I(t) represents current in amperes. From Kirchhoff’s laws, the algebraic
sum of the voltage drops is equal to the input voltage V (t). Then, Ohm’s Law implies
VR (t) + VC (t) = V (t) where VR (t) = RI(t) and Coulomb’s Law implies VC (t) = Q(t)
.
C
The charge Q(t) and the current I(t) satisfy the formula I(t) = Q0 (t) = CVC0 (t).
Reference: Edwards & Penney BVP3.7 and course WEB notes:
http://www.math.utah.edu/~gustafso/2250exp-modeling.pdf
http://www.math.utah.edu/~gustafso/s2014/2250/linearapplications2-5.pdf
Wikipedia reference for RC circuit: http://en.wikipedia.org/wiki/RC circuit
2. (30 points) Brine Tank Cascade
Consider two brine tanks with water cascading from tank 1 down into tank 2, and then
out of tank 2. See textbook Fig 1.5.5. Suppose that tank 1 initially contains 60 gallons
of brine, while tank 2 contains 120 gallons of pure water. When the system starts, fresh
water is pumped into tank 1 at a constant rate of 4 gal/min. At the same time, the brine
solution in tank 1 drains into tank 2 at a rate of 4 gal/min, and tank 2 drains at an equal
rate, causing the volumes in both tanks to remain constant. Assume that the solutions
in each tank remain well-mixed, so that although the salt concentrations are changing
in time, the concentration of salt leaving each tank equals the average concentration in
that tank.
(a) If the first tank originally contains 40 lbs of salt, then formulate and solve an
initial value problem to find the amount x(t) of salt in tank 1 at time t. Provide a
technology answer check.
=
(b) Suppose that y(t) is the amount of salt in tank 2 at time t. Show first that dy
dt
y
x
−
.
Then
solve
for
y(t)
using
the
function
x(t)
found
in
part
(a).
Provide
a
15
30
technology answer check.
Reference: Edwards & Penney Section 1.5 and Section 7.3, plus course WEB notes for
cascades and compartment analysis:
http://www.math.utah.edu/~gustafso/s2014/2250/systemsExamplesTheory2008.pdf
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3. (50 points) Paratrooper with Linear Drag
A paratrooper bails out of an airplane at the altitude of 15,000 ft, falls freely for 30
seconds, then opens a parachute. Assume linear air resistance ρv ft/s2 and take ρ=0.4
before the parachute opens and ρ= 4 after the parachute opens.
(a) Write down the ODE for velocity v of the paratrooper before the parachute opens.
(b) Find the velocity at time 5 seconds and the velocity at time 30 seconds, when the
paratrooper just opens the parachute. Provide a technology answer check.
(c) Find the height when the paratrooper opens the parachute.
(d) Explain why w0 = −32 − 4w, w(0) = v(30) ≈ −80 is the model for the paratrooper’s
descent after the parachute opens.
(e) Explain why the total time of descent for the paratrooper is t = 30 + 1598 = 1628s.
Provide a technology answer check.
Reference: Edwards & Penney Section 2.3. This traditional parachute problem can be
modified to be more realistic. Read Meade & Struthers (1999), Differential Equations
in the New Millennium: the Parachute Problem, Int. J. Engng Ed. 15(6), 417-424.
Retrieved on January 14, 2014 at
http://www.math.sc.edu/~meade/papers/ijee-parachute99.pdf.
See also Meade, D. (1998), ODE Models for the Parachute Problem, SIAM Review.
40(2). 327-332. Retrieved on January 14, 2014 at
http://epubs.siam.org/doi/pdf/10.1137/S0036144596316248.
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