Download Test 6 Review Test

Differential Equations Test
Word Problems
Name______________________
Solve the following.
1. An object of mass 3 kg is released from rest 500 meters above the ground
and allowed to fall under the influence of gravity (Use 9.81). The force due
to air resistance is proportional to the velocity of the object with
proportional constant k = 3 N-sec/m. Find the equation on the motion of the
object.
Answer: x(t) = 9.81t – 9.81(1-e-t) ;
2. The population of a town has double after 30 years. How many years
will be take before the population triples?
3. A 16-lb weight stretches a spring 8/3 ft. Initially the weight starts from
rest 2 ft below the equilibrium position and the subsequent motion takes
place in a medium that offers a damping force numerically equal to ½ dx/dt.
Find the equation of motion is the weight is driven by an external force
equal to f(t) = 10 cos 3t.
4. A 4 – lb weight is attached to a spring whose spring constant is 16 lb/ft.
What is the equation for the motion of the spring?
5. Forty percent of a substance has decayed after 10 years. How much of
the substance remains after 15 years?
6. Suppose that an 80-microgram sample of radioactive isotope decays to
70 micrograms after 10 hours. What is the half-life of the isotope (or how
long does it take for only half of the original sample to remain?
7. An 8-lb weight stretches a spring 2 ft. Assuming a damping force
numerically equal to two times the instantaneous velocity acts on the
system, determine the equation of motion if the weight is released from the
equilibrium position with an upward velocity of 3 ft/sec.
Partial answer: x(t) = c1e-4t + c2 te-4t
3. A circuit has in series an electromotive force of 5sin100t
V, a resistor of 10 ohms, an inductor of .25 henries and a capacitor of 4X10 -4
farads. If the initial current and the initial charge on the capacitor is both
zero, find the charge on the capacitor at any time t.