ecuaciones diferenciales

1. List No. 03
Problems proposed in the course of differential equations
Industrial Engineer
1. Velocity. We saw that the differential equation
m
dv
dt
= mg − kv
where k is a positive constant and g is the acceleration due to gravity, is a model for the
velocity v of a body of mass m that is falling under the influent of gravity. Because the term
−kv represents air resistance, the velocity of a body falling from a great height does not increase
without bound as time t increases.
Find the speed v(t), speed limit vlimite and speed v(t1)
a) m = 100 kg, g = 9.8 m/seg2
, b = 5 kg/seg, v(0) = 50 m/seg, find v(3)
b) m = 40 kg, g = 9.8 m/seg2
, b = 10 kg/seg, v(0) = 60 m/seg, find v(2)
Figura 1: Terminal velocity
2. Velocity. Suppose the model in the previous problem is modified so that air resistance is
proportional to v2
, that is,
m
dv
dt
= mg − kv2
Find the solution to v(t).
3. Velocity. An object of mass 5 kg is released from rest 1000 m above the ground and allowed
to fall under the influence of gravity. Assuming the force due to air resistance is proportional
to the velocity of the object with proportionality constant b = 50 N − sec/m, determine the
equation of motion of the object. When will the object strike the ground?
Luis Lara Romero. c Copyright All rights reserved1

2. 4. Velocity. A 400 lb object is released from rest 500 ft above the ground and allowed to fall
under the influence of gravity. Assuming that the force in pounds due to air resistance is −10v,
where v is the velocity of the object in ft/sec, determine the equation of motion of the object.
When will the object hit the ground?
5. Velocity. If the object in Problem (3) has a mass of 500 kg instead of 5 kg, when will it strike
the ground ? [Hill: Here the exponential term is too large to ignore. Use Newton’s method to
approximate the time t when the object strikes the ground].
6. Velocity. An object of mass 5 kg is given an initial downward velocity of 50 m/sec and then
allowed to fall under the influence of gravity. Assume that the force in Newton due to air
resistance is −10v, where v is the velocity of the object in m/sec. If the object is initially 500
m above the ground, determine when the object will strike the ground.
7. Velocity. A parachutist whose mass is 75 kg drops from a helicopter hovering 2000 m above
the ground and falls toward the ground under the influence of gravity. Assume that the force
due to air resistance is proportional to the velocity of the parachutist, with the proportionality
constant b1 = 30 N − sec/m when the chute is closed and b2 = 90 N − sec/m when the chute
is open. If the chute does not open until the velocity of the parachutist reaches 20 m/sec, after
how many seconds will she reach the ground?.
8. Growth and Decay. The population of a town grows at a rate proportional to the population
present at time t. The initial population of 500 increases by 15 % in 10 years. What will be the
population in 30 years?. How fast is the population growing at t = 30 ?.
9. Growth and Decay. The population of bacteria in a culture grows at a rate proportional to
the number of bacteria present at time t. After 3 hours it is observed that 400 bacteria are
present. After 10 hours 2000 bacteria are present. What was the initial number of bacteria?.
10. Growth and Decay. The radioactive isotope of lead, Pb − 209, decays at a rate proportional
to the amount present at time t and has a half-life of 3.3 hours. If 1 gram of this isotope is
present initially, how long will it take for 90 % of the lead to decay?
11. Growth and Decay. Initially 100 milligrams of a radioactive substance was present. After 6
hours the mass had decreased by 3 %. If the rate of decay is proportional to the amount of the
substance present at time t, find the amount remaining after 24 hours.
12. Growth and Decay. Determine the half-life of the radioactive substance described in Problem
11.
Luis Lara Romero. c Copyright All rights reserved2