Mathematical Models and Numerical Methods Involving First-Order Equations

An object at rest on an inclined plane will not slide until the component of the gravitational force down the incline is sufficient to overcome the force due to static friction. Static friction is governed by an experimental law somewhat like that of kinetic friction (Problem 18); it has a magnitude of at most N, where m is the coefficient of static friction and N is, again, the magnitude of the normal force exerted by the surface on the object. If the plane is inclined at an angle a, determine the critical value ₀ for which the object will slide if $\mathit{a}\mathbf{>}\mathit{a}$_o   but will not move for $\mathit{a}\mathbf{<}\mathit{a}$_o.

Use the fourth-order Runge–Kutta subroutine with h = 0.25 to approximate the solution to the initial value problem $\mathbf{y}\mathbf{\text{'}}\mathbf{=}\mathbf{2}\mathbf{y}\mathbf{-}\mathbf{6}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}$, at x = 1. (Thus, input N = 4.) Compare this approximation to the actual solution $\mathbf{y}\mathbf{=}\mathbf{3}\mathbf{-}\mathbf{2}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}$ evaluated at x = 1.

A swimming pool whose volume is 10,000 gal contains water that is 0.01% chlorine. Starting at t = 0, city water containing 0.001% chlorine is pumped into the pool at a rate of 5 gal/min. The pool water flows out at the same rate. What is the percentage of chlorine in the pool after 1 h? When will the pool water be 0.002% chlorine?

The air in a small room 12 ft by 8 ft by 8 ft is 3% carbon monoxide. Starting at t = 0, fresh air containing no carbon monoxide is blown into the room at a rate of $\mathbf{100}\mathit{f}{\mathit{t}}^{\mathbf{3}}\mathbf{/}\mathit{m}\mathit{i}\mathit{n}$. If air in the room flows out through a vent at the same rate, when will the air in the room be 0.01% carbon monoxide?

Beginning at time t=0, fresh water is pumped at the rate of 3 gal/min into a 60-gal tank initially filled with brine. The resulting less-and-less salty mixture overflows at the same rate into a second 60-gal tank that initially contained only pure water, and from there it eventually spills onto the ground. Assuming perfect mixing in both tanks, when will the water in the second tank taste saltiest? And exactly how salty will it then be, compared with the original brine?

In 1990 the Department of Natural Resources released 1000 splake (a crossbreed of fish) into a lake. In 1997 the population of splake in the lake was estimated to be 3000. Using the Malthusian law for population growth, estimate the population of splake in the lake in the year 2020.

In Problem 9, suppose we have the additional information that the population of splake in 2004 was estimated to be 5000. Use a logistic model to estimate the population of splake in the year 2020. What is the predicted limiting population? [Hint: Use the formulas in Problem 12.

In 1980 the population of alligators on the Kennedy Space Center grounds was estimated to be 1500. In 2006 the population had grown to an estimated 6000. Using the Malthusian law for population growth, estimate the alligator population on the Kennedy Space Center grounds in the year 2020.

In Problem 14, suppose we have the additional information that the population of alligators on the grounds of the Kennedy Space Center in 1993 was estimated to be 4100. Use a logistic model to estimate the population of alligators in the year 2020. What is the predicted limiting population? [Hint: Use the formulas in Problem 12.

From theoretical considerations, it is known that light from a certain star should reach Earth with intensity l₀ . However, the path taken by the light from the star to Earth passes through a dust cloud, with absorption coefficient 0.1/light-year. The light reaching Earth has intensity 1/2 l₀. How thick is the dust cloud? (The rate of change of light intensity with respect to thickness is proportional to the intensity. One light-year is the distance travelled by light during 1 yr.)

A snowball melts in such a way that the rate of change in its volume is proportional to its surface area. If the snowball was initially 4 in. in diameter and after 30 min its diameter is 3 in., when will its diameter be 2 in.? Mathematically speaking, when will the snowball disappear?

Suppose the snowball in Problem 21 melts so that the rate of change in its diameter is proportional to its surface area. Using the same given data, determine when its diameter will be 2 in. Mathematically speaking, when will the snowball disappear?

In Problems 23–27, assume that the rate of decay of a radioactive substance is proportional to the amount of the substance present. The half-life of a radioactive substance is the time it takes for one-half of the substance to disintegrate. If initially there are 300 g of a radioactive substance and after 5 yr there are 200 g remaining, how much time must elapse before only 10 g remain?

In Problems 23–27, assume that the rate of decay of a radioactive substance is proportional to the amount of the substance present. The half-life of a radioactive substance is the time it takes for one-half of the substance to disintegrate. If initially there are 50 g of a radioactive substance and after 3 days there are only 10 g remaining, what percentage of the original amount remains after 4 days?

In Problems 23–27, assume that the rate of decay of a radioactive substance is proportional to the amount of the substance present. The half-life of a radioactive substance is the time it takes for one-half of the substance to disintegrate.

To see how sensitive the technique of carbon dating of Problem 25 is

(a) Redo Problem 25 assuming the half-life of carbon-14 is 5550 yr.

(b) Redo Problem 25 assuming 3% of the original mass remains.

In Problems 23–27, assume that the rate of decay of a radioactive substance is proportional to the amount of the substance present. The half-life of a radioactive substance is the time it takes for one-half of the substance to disintegrate. Carbon dating is often used to determine the age of a fossil. For example, a humanoid skull was found in a cave in South Africa along with the remains of a campfire. Archaeologists believe the age of the skull to be the same age as the campfire. It is determined that only 2% of the original amount of carbon-14 remains in the burnt wood of the campfire. Estimate the age of the skull if the half-life of carbon-14 is about 5600 years.

A red wine is brought up from the wine cellar, which is a cool 10°C, and left to breathe in a room of temperature 23°C. If it takes 10 min for the wine to reach 15°C, when will the temperature of the wine reach 18°C?

A white wine at room temperature 70°F is chilled in ice (32°F). If it takes 15 min for the wine to chill to 60°F, how long will it take for the wine to reach 56°F?

A cold beer initially at 35°F warms up to 40°F in 3 min while sitting in a room of temperature 70°F. How warm will the beer be if left out for 20 min?

A cup of hot coffee initially at 95°C cools to 80°C in 5 min while sitting in a room of temperature 21°C. Using just Newton’s law of cooling, determine when the temperature of the coffee will be a nice 50°C.

In Problem 13, if a larger tank with a heat capacity of $\mathbf{1}\mathbf{°}\mathbf{F}$ per thousand Btu and a time constant of 72 hr is use instead (with all other factors being the same), what will be the temperature in the tank after 12 hr?

A solar hot-water-heating system consists of a hot-water tank and a solar panel. The tank is well insulated and has a time constant of 64 hr. The solar panel generates 2000 Btu/hr during the day, and the tank has a heat capacity of $\mathbf{2}\mathbf{°}\mathbf{F}\mathbf{ }$ per thousand Btu. If the water in the tank is initially  $\mathbf{80}\mathbf{°}\mathbf{F}$and the room temperature outside the tank is , what will be the temperature in the tank after 12 hr of sunlight?

Two friends sit down to talk and enjoy a cup of coffee. When the coffee is served, the impatient friend immediately adds a teaspoon of cream to his coffee. The relaxed friend waits 5 min before adding a teaspoon of cream (which has been kept at a constant temperature). The two now begin to drink their coffee. Who has the hotter coffee? Assume that the cream is cooler than the air and has the same heat capacity per unit volume as the coffee, and that Newton’s law of cooling governs the heat transfer.

During the summer the temperature inside a van reaches $\mathbf{55}\mathbf{°}\mathbf{C}$, while that outside is a constant $\mathbf{35}\mathbf{°}\mathbf{C}$. When the driver gets into the van, she turns on the air conditioner with the thermostat set at $\mathbf{16}\mathbf{°}\mathbf{C}$. If the time constant for the van is $\frac{\mathbf{1}}{\mathbf{k}}\mathbf{=}\mathbf{2} \mathbf{hr}$ and that for the van with its air conditioning system is $\frac{\mathbf{1}}{{\mathbf{k}}_{\mathbf{1}}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}} \mathbf{hr}$, when will the temperature inside the van reach  $\mathbf{27}\mathbf{°}\mathbf{C}$?

Early Monday morning, the temperature in the lecture hall has fallen to $\mathbf{40}\mathbf{°}\mathbf{F}$, the same as the temperature outside. At $\mathbf{7}\mathbf{:}\mathbf{00}\mathbf{a}\mathbf{.}\mathbf{m}\mathbf{.}$, the janitor turns on the furnace with the thermostat set at $\mathbf{70}\mathbf{°}\mathbf{F}$. The time constant for the building is $\frac{\mathbf{1}}{\mathbf{k}}\mathbf{=}\mathbf{2}\mathbf{ }\mathit{h}\mathit{r}$ and that for the building along with its heating system is $\frac{\mathbf{1}}{{\mathbf{k}}_{\mathbf{1}}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{ }\mathit{h}\mathit{r}$. Assuming that the outside temperature remains constant, what will be the temperature inside the lecture hall at $\mathbf{8}\mathbf{:}\mathbf{00}\mathbf{a}\mathbf{.}\mathbf{m}\mathbf{.}$? When will the temperature inside the hall reach $\mathbf{65}\mathbf{°}\mathbf{F}$?

A warehouse is being built that will have neither heating nor cooling. Depending on the amount of insulation, the time constant for the building may range from 1 to 5 hr. To illustrate the effect insulation will have on the temperature inside the warehouse, assume the outside temperature varies as a sine wave, with a minimum of $\mathbf{16}\mathbf{°}\mathbf{C}$ at $\mathbf{2}\mathbf{:}\mathbf{00}\mathbf{a}\mathbf{.}\mathbf{m}\mathbf{.}$ and a maximum of $\mathbf{32}\mathbf{°}\mathbf{C}$ at $\mathbf{2}\mathbf{:}\mathbf{00}\mathbf{p}\mathbf{.}\mathbf{m}\mathbf{.}$ Assuming the exponential term (which involves the initial temperature T0) has died off, what is the lowest temperature inside the building if the time constant is 1 hr? If it is 5 hr? What is the highest temperature inside the building if the time constant is 1 hr? If it is 5 hr?

A garage with no heating or cooling has a time constant of 2 hr. If the outside temperature varies as a sine wave with a minimum of $\mathbf{50}\mathbf{°}\mathbf{F}$ at $\mathbf{2}\mathbf{:}\mathbf{00}\mathbf{a}\mathbf{.}\mathbf{m}\mathbf{.}$ and a maximum of  $\mathbf{80}\mathbf{°}\mathbf{F}$at $\mathbf{2}\mathbf{:}\mathbf{00}\mathbf{p}\mathbf{.}\mathbf{m}\mathbf{.}$, determine the times at which the building reaches its lowest temperature and its highest temperature, assuming the exponential term has died off.

On a hot Saturday morning while people are working inside, the air conditioner keeps the temperature inside the building at $\mathbf{24}\mathbf{°}\mathbf{C}$. At noon the air conditioner is turned off, and the people go home. The temperature outside is a constant $\mathbf{35}\mathbf{°}\mathbf{C}$ for the rest of the afternoon. If the time constant for the building is 4 hr, what will be the temperature inside the building at 2:00 p.m.? At 6:00 p.m.? When will the temperature inside the building reach $\mathbf{27}\mathbf{°}\mathbf{C}$?

On a mild Saturday morning while people are working inside, the furnace keeps the temperature inside the building at 21°C. At noon the furnace is turned off, and the people go home. The temperature outside is a constant 12°C for the rest of the afternoon. If the time constant for the building is 3 hr, when will the temperature inside the building reach 16°C? If some windows are left open and the time constant drops to 2 hr, when will the temperature inside reach 16°C?

It was noon on a cold December day in Tampa: 16°C. Detective Taylor arrived at the crime scene to find the sergeant leaning over the body. The sergeant said there were several suspects. If they knew the exact time of death, then they could narrow the list. Detective Taylor took out a thermometer and measured the temperature of the body: 34.5°C. He then left for lunch. Upon returning at 1:00 p.m., he found the body temperature to be 33.7°C. When did the murder occur? [Hint: Normal body temperature is 37°C.]

In Problems 23–27, assume that the rate of decay of a radioactive substance is proportional to the amount of the substance present. The half-life of a radioactive substance is the time it takes for one-half of the substance to disintegrate. The only undiscovered isotopes of the two unknown elements hohum and inertium (symbols Hh and It) are radioactive. Hohum decays into inertium with a decay constant of 2/yr, and inertium decays into the nonradioactive isotope of bunkum (symbol Bu) with a decay constant of 1/yr. An initial mass of 1 kg of hohum is put into a non-radiaoctive container, with no other source of hohum, inertium, or bunkum. How much of each of the three elements is in the container after t yr? (The decay constant is the constant of proportionality in the statement that the rate of loss of mass of the element at any time is proportional to the mass of the element at that time.)

If the object in Problem 1 has a mass of  500 kg instead of 5 kg , when will it strike the ground? [Hint: Here the exponential term is too large to ignore. Use Newton’s method to approximate the time t when the object strikes the ground (see Appendix B)

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?

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?

If the object in Problem 2 is released from rest 30ft above the ground instead of 500ft, when will it strike the ground? [ Hint: Use Newton’s method to solve for t ]

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 newtons due to air resistance is -10v, where v is the velocity of the object in m/sec. Determine the equation of motion of the object. If the object is initially 500 m above the ground, determine when the object will strike the ground.

An object of mass 8 kg  is given an upward initial velocity of 20 m/sec and then allowed to fall under the influence of gravity. Assume that the force in newtons due to air resistance is -16v , where v is the velocity of the object in m/sec. Determine the equation of motion of the object. If the object is initially 100 m above the ground, determine when the object will strike the ground.

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?

A parachutist whose mass is 100 kg drops from a helicopter hovering 3000 m above the ground and falls 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 b₃=20 N-sec/m when the chute is closed and  b₄=100 N-sec/m when the chute is open. If the chute does not open until 30 sec after the parachutist leaves the helicopter, after how many seconds will he hit the ground? If the chute does not open until 1 min after he leaves the helicopter, after how many seconds will he hit the ground?

An object of mass m is released from rest and falls under the influence of gravity. If the magnitude of the force due to air resistance is bvⁿ, where b and n are positive constants, find the limiting velocity of the object (assuming this limit exists). [Hint: Argue that the existence of a (finite) limiting velocity implies that  $\frac{\mathbf{dv}}{\mathbf{dt}}\mathbf{\to }\mathbf{0}$ as  $\mathbf{\text{t}}\mathbf{\to }\mathbf{+}\mathbf{\infty }$

A rotating flywheel is being turned by a motor that exerts a constant torque T (see Figure 3.10). A retarding torque due to friction is proportional to the angular velocity v. If the moment of inertia of the flywheel, is I and its initial angular velocity is , find the equation for the angular velocity v as a function of time. [Hint: Use Newton’s second law for rotational motion, that is, moment of inertia * angular acceleration = sum of the torques.]

Find the equation for the angular velocity $\mathit{\omega }$  in Problem15, assuming that the retarding torque is proportional to   $\sqrt{\mathbf{\omega }}$

In Problem 16, let I = 50 kg-m² and the retarding torque be    N-mIf the motor is turned off with the angular velocity at 225 rad/sec, determine how long it will take for the flywheel to come to rest.

When an object slides on a surface, it encounters a resistance force called friction. This force has a magnitude of $\mathbf{\mu N}$ , where $\mathbf{\mu }$  the coefficient of kinetic friction and N is the magnitude of the normal force that the surface applies to the object. Suppose an object of mass  30 kg is released from the top of an inclined plane that is inclined  $\mathbf{30}\mathbf{°}$ to the horizontal (see Figure 3.11). Assume the gravitational force is constant, air resistance is negligible, and the coefficient of kinetic friction $\mathbf{\mu }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{2}$ . Determine the equation of motion for the object as it slides down the plane. If the top surface of the plane is 5 m long, what is the velocity of the object when it reaches the bottom?

An object of mass 60 kg starts from rest at the top of a 45º inclined plane. Assume that the coefficient of kinetic friction is 0.05 (see Problem 18). If the force due to air resistance is proportional to the velocity of the object, say, -3, find the equation of motion of the object. How long will it take the object to reach the bottom of the inclined plane if the incline is 10 m long?

An object of mass 100kg  is released from rest from a boat into the water and allowed to sink. While gravity is pulling the object down, a buoyancy force of 1/40  times the weight of the object is pushing the object up (weight = mg). If we assume that water resistance exerts a force on the object that is proportional to the velocity of the object, with proportionality constant 10N-sec/m , find the equation of motion of the object. After how many seconds will the velocity of the object be 70 m/sec ?

An object of mass 2 kg is released from rest from a platform 30 m above the water and allowed to fall under the influence of gravity. After the object strikes the water, it begins to sink with gravity pulling down and a buoyancy force pushing up. Assume that the force of gravity is constant, no change in momentum occurs on impact with the water, the buoyancy force is 1/2 the weight (weight = mg), and the force due to air resistance or water resistance is proportional to the velocity, with proportionality constant   b₁ = 10 N-sec/m in the air and   b₂= 100 N-sec/m in the water. Find the equation of motion of the object. What is the velocity of the object 1 min after it is released?

In Example 1, we solved for the velocity of the object as a function of time (equation (5)). In some cases, it is useful to have an expression, independent of t, that relates v and x. Find this relation for the motion in Example 1. [Hint: Letting $\mathbf{v}\left(t\right)\mathbf{=}\mathbf{V}\left(x\left(t\right)\right)$, then $\frac{\mathbf{d}\mathbf{v}}{\mathbf{d}\mathbf{t}}\mathbf{=}\left(\frac{dv}{dx}\right)\mathbf{V}$]