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60 Chapter 2. First Order Differential Equations
Solving Eq. (31) for v0, we find the initial velocity required to lift the body to the altitude ξ, namely,
v0 = √
2gR ξ
R + ξ . (32)
The escape velocity ve is then found by letting ξ → ∞. Consequently,
ve = √
2gR. (33)
The numerical value of ve is approximately 6.9 mi/s, or 11.1 km/s. The preceding calculation of the escape velocity neglects the effect of air resistance, so the
actual escape velocity (including the effect of air resistance) is somewhat higher. On the other hand, the effective escape velocity can be significantly reduced if the body is transported a considerable distance above sea level before being launched. Both gravitational and frictional forces are thereby reduced;air resistance, in particular,diminishes quite rapidly with increasing altitude. You should keep in mind also that it may well be impractical to impart too large an initial velocity instantaneously; space vehicles, for instance, receive their initial acceleration during a period of a few minutes.
PROBLEMS 1. Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains 200 L of a dye solution with a concentration of 1 g/L. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of 2 L/min, the well-stirred solution flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank reaches 1% of its original value.
2. A tank initially contains 120 L of pure water. A mixture containing a concentration of γ g/L of salt enters the tank at a rate of 2 L/min, and the well-stirred mixture leaves the tank at the same rate. Find an expression in terms of γ for the amount of salt in the tank at any time t. Also find the limiting amount of salt in the tank as t → ∞.
3. A tank originally contains 100 gal of fresh water. Then water containing 12 lb of salt per gallon is poured into the tank at a rate of 2 gal/min, and the mixture is allowed to leave at the same rate. After 10 min the process is stopped, and fresh water is poured into the tank at a rate of 2 gal/min, with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional 10 min.
4. A tank with a capacity of 500 gal originally contains 200 gal of water with 100 lb of salt in solution. Water containing 1 lb of salt per gallon is entering at a rate of 3 gal/min, and the mixture is allowed to flow out of the tank at a rate of 2 gal/min. Find the amount of salt in the tank at any time prior to the instant when the solution begins to overflow. Find the concentration (in pounds per gallon) of salt in the tank when it is on the point of overflowing. Compare this concentration with the theoretical limiting concentration if the tank had infinite capacity.
5. A tank contains 100 gal of water and 50 oz of salt. Water containing a salt concentration of 1 4 (1 + 12 sin t) oz/gal flows into the tank at a rate of 2 gal/min, and the mixture in the tank flows out at the same rate. (a) Find the amount of salt in the tank at any time. (b) Plot the solution for a time period long enough so that you see the ultimate behavior of the graph. (c) The long-time behavior of the solution is an oscillation about a certain constant level. What is this level? What is the amplitude of the oscillation?
HE Mr Select exercisesIn RED tea
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2.3 Modeling with First Order Equations 61
6. Suppose that a tank containing a certain liquid has an outlet near the bottom. Let h(t) be the height of the liquid surface above the outlet at time t. Torricelli’s2 principle states that the outflow velocity v at the outlet is equal to the velocity of a particle falling freely (with no drag) from the height h. (a) Show that v =
√ 2gh, where g is the acceleration due to gravity.
(b) By equating the rate of outflow to the rate of change of liquid in the tank, show that h(t) satisfies the equation
A(h) dh dt
= −αa √
2gh, (i)
where A(h) is the area of the cross section of the tank at height h and a is the area of the outlet. The constant α is a contraction coefficient that accounts for the observed fact that the cross section of the (smooth) outflow stream is smaller than a. The value of α for water is about 0.6. (c) Consider a water tank in the form of a right circular cylinder that is 3 m high above the outlet. The radius of the tank is 1 m, and the radius of the circular outlet is 0.1 m. If the tank is initially full of water, determine how long it takes to drain the tank down to the level of the outlet.
7. Suppose that a sum S0 is invested at an annual rate of return r compounded continuously. (a) Find the time T required for the original sum to double in value as a function of r. (b) Determine T if r = 7%. (c) Find the return rate that must be achieved if the initial investment is to double in 8 years.
8. A young person with no initial capital invests k dollars per year at an annual rate of return r. Assume that investments are made continuously and that the return is compounded continuously. (a) Determine the sum S(t) accumulated at any time t. (b) If r = 7.5%, determine k so that $1 million will be available for retirement in 40 years. (c) If k = $2000/year,determine the return rate r that must be obtained to have $1 million available in 40 years.
9. A certain college graduate borrows $8000 to buy a car. The lender charges interest at an annual rate of 10%. Assuming that interest is compounded continuously and that the borrower makes payments continuously at a constant annual rate k, determine the payment rate k that is required to pay off the loan in 3 years. Also determine how much interest is paid during the 3-year period.
10. A home buyer can afford to spend no more than $1500/month on mortgage payments. Suppose that the interest rate is 6%, that interest is compounded continuously, and that payments are also made continuously. (a) Determine the maximum amount that this buyer can afford to borrow on a 20-year mortgage; on a 30-year mortgage. (b) Determine the total interest paid during the term of the mortgage in each of the cases in part (a).
2Evangelista Torricelli (1608–1647), successor to Galileo as court mathematician in Florence, published this result in 1644. He is also known for constructing the first mercury barometer and for making important contributions to geometry.
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11. A home buyer wishes to borrow $250,000 at an interest rate of 6% to finance the pur- chase. Assume that interest is compounded continuously and that payments are also made continuously. (a) Determine the monthly payment that is required to pay off the loan in 20 years; in 30 years. (b) Determine the total interest paid during the term of the loan in each of the cases in part (a).
12. A recent college graduate borrows $150,000 at an interest rate of 6% to purchase a con- dominium. Anticipating steady salary increases, the buyer expects to make payments at a monthly rate of 800 + 10t, where t is the number of months since the loan was made. (a) Assuming that this payment schedule can be maintained, when will the loan be fully paid? (b) Assuming the same payment schedule, how large a loan could be paid off in exactly 20 years?
13. An important tool in archeological research is radiocarbon dating, developed by the American chemistWillard F. Libby.3 This is a means of determining the age of certain wood and plant remains, and hence of animal or human bones or artifacts found buried at the same levels. Radiocarbon dating is based on the fact that some wood or plant remains con- tain residual amounts of carbon-14,a radioactive isotope of carbon.This isotope is accumu- lated during the lifetime of the plant and begins to decay at its death. Since the half-life of carbon-14 is long (approximately 5730 years4), measurable amounts of carbon-14 remain after many thousands of years. If even a tiny fraction of the original amount of carbon-14 is still present, then by appropriate laboratory measurements the proportion of the orig- inal amount of carbon-14 that remains can be accurately determined. In other words, if Q(t) is the amount of carbon-14 at time t and Q0 is the original amount, then the ratio Q(t)/Q0 can be determined, as long as this quantity is not too small. Present measurement techniques permit the use of this method for time periods of 50,000 years or more. (a) Assuming that Q satisfies the differential equation Q′ = −rQ, determine the decay constant r for carbon-14. (b) Find an expression for Q(t) at any time t, if Q(0) = Q0. (c) Suppose that certain remains are discovered in which the current residual amount of carbon-14 is 20% of the original amount. Determine the age of these remains.
14. Suppose that a certain population has a growth rate that varies with time and that this population satisfies the differential equation
dy/dt = (0.5 + sin t)y/5.
(a) If y(0) = 1, find (or estimate) the time τ at which the population has doubled. Choose other initial conditions and determine whether the doubling time τ depends on the initial population. (b) Suppose that the growth rate is replaced by its average value 1/10. Determine the doubling time τ in this case.
3Willard F. Libby (1908–1980) was born in rural Colorado and received his education at the University of California at Berkeley. He developed the method of radiocarbon dating beginning in 1947 while he was at the University of Chicago. For this work he was awarded the Nobel Prize in chemistry in 1960. 4McGraw-Hill Encyclopedia of Science and Technology (8th ed.) (New York: McGraw-Hill, 1997), Vol. 5, p. 48.
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(c) Suppose that the term sin t in the differential equation is replaced by sin 2πt; that is, the variation in the growth rate has a substantially higher frequency. What effect does this have on the doubling time τ? (d) Plot the solutions obtained in parts (a), (b), and (c) on a single set of axes.
15. Suppose that a certain population satisfies the initial value problem
dy/dt = r(t)y −k, y(0) = y0,
where the growth rate r(t) is given by r(t) = (1 + sin t)/5, and k represents the rate of predation. (a) Suppose that k = 1/5. Plot y versus t for several values of y0 between 1/2 and 1. (b) Estimate the critical initial population yc below which the population will become extinct. (c) Choose other values of k and find the corresponding yc for each one. (d) Use the data you have found in parts (b) and (c) to plot yc versus k.
16. Newton’s law of cooling states that the temperature of an object changes at a rate pro- portional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton’s law of cooling. If the coffee has a temperature of 200◦F when freshly poured, and 1 min later has cooled to 190◦F in a room at 70◦F, determine when the coffee reaches a temperature of 150◦F.
17. Heat transfer from a body to its surroundings by radiation, based on the Stefan– Boltzmann5 law, is described by the differential equation
du dt
= −α(u4 −T4), (i)
where u(t) is the absolute temperature of the body at time t, T is the absolute temperature of the surroundings, and α is a constant depending on the physical parameters of the body. However, if u is much larger than T , then solutions of Eq. (i) are well approximated by solutions of the simpler equation
du dt
= −αu4. (ii)
Suppose that a body with initial temperature 2000 K is surrounded by a medium with temperature 300 K and that α = 2.0 × 10−12 K−3/s. (a) Determine the temperature of the body at any time by solving Eq. (ii). (b) Plot the graph of u versus t. (c) Find the time τ at which u(τ) = 600—that is, twice the ambient temperature. Up to this time the error in using Eq. (ii) to approximate the solutions of Eq. (i) is no more than 1%.
18. Consider an insulated box (a building, perhaps) with internal temperature u(t). According to Newton’s law of cooling, u satisfies the differential equation
du dt
= −k[u −T(t)], (i)
where T(t) is the ambient (external) temperature. Suppose that T(t) varies sinusoidally; for example, assume that T(t) = T0 + T1 cos ωt.
5Jozef Stefan (1835–1893), professor of physics at Vienna, stated the radiation law on empirical grounds in 1879. His student Ludwig Boltzmann (1844–1906) derived it theoretically from the principles of thermodynamics in 1884. Boltzmann is best known for his pioneering work in statistical mechanics.
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64 Chapter 2. First Order Differential Equations
(a) Solve Eq. (i) and express u(t) in terms of t, k, T0, T1, and ω. Observe that part of your solution approaches zero as t becomes large; this is called the transient part. The remainder of the solution is called the steady state; denote it by S(t). (b) Suppose that t is measured in hours and that ω = π/12, corresponding to a period of 24 h for T(t). Further, let T0 = 60◦F, T1 = 15◦F, and k = 0.2/h. Draw graphs of S(t) and T(t) versus t on the same axes. From your graph estimate the amplitude R of the oscillatory part of S(t). Also estimate the time lag τ between corresponding maxima of T(t) and S(t). (c) Let k, T0, T1, and ω now be unspecified. Write the oscillatory part of S(t) in the form R cos[ω(t −τ)]. Use trigonometric identities to find expressions for R and τ. Let T1 and ω have the values given in part (b), and plot graphs of R and τ versus k.
19. Consider a lake of constant volume V containing at time t an amount Q(t) of pollutant, evenly distributed throughout the lake with a concentration c(t), where c(t) = Q(t)/V . Assume that water containing a concentration k of pollutant enters the lake at a rate r, and that water leaves the lake at the same rate. Suppose that pollutants are also added directly to the lake at a constant rate P. Note that the given assumptions neglect a number of factors that may, in some cases, be important—for example, the water added or lost by precipitation, absorption, and evaporation; the stratifying effect of temperature differ- ences in a deep lake; the tendency of irregularities in the coastline to produce sheltered bays; and the fact that pollutants are deposited unevenly throughout the lake but (usually) at isolated points around its periphery. The results below must be interpreted in the light of the neglect of such factors as these. (a) If at time t = 0 the concentration of pollutant is c0, find an expression for the concentration c(t) at any time. What is the limiting concentration as t → ∞? (b) If the addition of pollutants to the lake is terminated (k = 0 and P = 0 for t > 0), determine the time interval T that must elapse before the concentration of pollutants is reduced to 50% of its original value; to 10% of its original value. (c) Table 2.3.2 contains data6 for several of the Great Lakes. Using these data, determine from part (b) the time T that is needed to reduce the contamination of each of these lakes to 10% of the original value.
TABLE 2.3.2 Volume and Flow Data for the Great Lakes
Lake V (km3 × 103) r (km3/year) Superior 12.2 65.2 Michigan 4.9 158 Erie 0.46 175 Ontario 1.6 209
20. A ball with mass 0.15 kg is thrown upward with initial velocity 20 m/s from the roof of a building 30 m high. Neglect air resistance. (a) Find the maximum height above the ground that the ball reaches. (b) Assuming that the ball misses the building on the way down, find the time that it hits the ground. (c) Plot the graphs of velocity and position versus time.
6This problem is based on R. H. Rainey, “Natural Displacement of Pollution from the Great Lakes,” Science 155 (1967), pp. 1242–1243; the information in the table was taken from that source.
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21. Assume that the conditions are as in Problem 20 except that there is a force due to air resistance of magnitude |v|/30 directed opposite to the velocity, where the velocity v is measured in m/s. (a) Find the maximum height above the ground that the ball reaches. (b) Find the time that the ball hits the ground. (c) Plot the graphs of velocity and position versus time. Compare these graphs with the corresponding ones in Problem 20.
22. Assume that the conditions are as in Problem 20 except that there is a force due to air resistance of magnitude v2/1325 directed opposite to the velocity, where the velocity v is measured in m/s. (a) Find the maximum height above the ground that the ball reaches. (b) Find the time that the ball hits the ground. (c) Plot the graphs of velocity and position versus time. Compare these graphs with the corresponding ones in Problems 20 and 21.
23. A skydiver weighing 180 lb (including equipment) falls vertically downward from an alti- tude of 5000 ft and opens the parachute after 10 s of free fall. Assume that the force of air resistance, which is directed opposite to the velocity, is of magnitude 0.75|v| when the parachute is closed and is of magnitude 12|v| when the parachute is open, where the velocity v is measured in ft/s. (a) Find the speed of the skydiver when the parachute opens. (b) Find the distance fallen before the parachute opens. (c) What is the limiting velocity vL after the parachute opens? (d) Determine how long the sky diver is in the air after the parachute opens. (e) Plot the graph of velocity versus time from the beginning of the fall until the skydiver reaches the ground.
24. A rocket sled having an initial speed of 150 mi/h is slowed by a channel of water. Assume that during the braking process, the acceleration a is given by a(v) = −µv2, where v is the velocity and µ is a constant. (a) As in Example 4 in the text, use the relation dv/dt = v(dv/dx) to write the equation of motion in terms of v and x. (b) If it requires a distance of 2000 ft to slow the sled to 15 mi/h, determine the value of µ. (c) Find the time τ required to slow the sled to 15 mi/h.
25. A body of constant mass m is projected vertically upward with an initial velocity v0 in a medium offering a resistance k|v|, where k is a constant. Neglect changes in the gravitational force. (a) Find the maximum height xm attained by the body and the time tm at which this maximum height is reached. (b) Show that if kv0/mg < 1, then tm and xm can be expressed as tm = v0 g [ 1 −1 2 kv0 mg + 1 3 ( kv0 mg )2 −· · · ] , xm = v20 2g [ 1 −2 3 kv0 mg + 1 2 ( kv0 mg )2 −· · · ] . (c) Show that the quantity kv0/mg is dimensionless. 26. A body of mass m is projected vertically upward with an initial velocity v0 in a medium offering a resistance k|v|, where k is a constant. Assume that the gravitational attraction of the earth is constant. (a) Find the velocity v(t) of the body at any time. • • August 7, 2012 21:03 c02 Sheet number 36 Page number 66 cyan black 66 Chapter 2. First Order Differential Equations (b) Use the result of part (a) to calculate the limit of v(t) as k → 0—that is, as the resis- tance approaches zero. Does this result agree with the velocity of a mass m projected upward with an initial velocity v0 in a vacuum? (c) Use the result of part (a) to calculate the limit of v(t) as m → 0—that is, as the mass approaches zero. 27. A body falling in a relatively dense fluid, oil for example, is acted on by three forces (see Figure 2.3.5): a resistive force R, a buoyant force B, and its weight w due to gravity. The buoyant force is equal to the weight of the fluid displaced by the object. For a slowly moving spherical body of radius a, the resistive force is given by Stokes’s law, R = 6πµa|v|, where v is the velocity of the body, and µ is the coefficient of viscosity of the surrounding fluid.7 (a) Find the limiting velocity of a solid sphere of radius a and density ρ falling freely in a medium of density ρ′ and coefficient of viscosity µ. (b) In 1910 R. A. Millikan8 studied the motion of tiny droplets of oil falling in an electric field. A field of strength E exerts a force Ee on a droplet with charge e. Assume that E has been adjusted so the droplet is held stationary (v = 0) and that w and B are as given above. Find an expression for e. Millikan repeated this experiment many times, and from the data that he gathered he was able to deduce the charge on an electron. R w a B FIGURE 2.3.5 A body falling in a dense fluid. 28. A mass of 0.25 kg is dropped from rest in a medium offering a resistance of 0.2|v|, where v is measured in m/s. (a) If the mass is dropped from a height of 30 m, find its velocity when it hits the ground. (b) If the mass is to attain a velocity of no more than 10 m/s, find the maximum height from which it can be dropped. 7Sir George Gabriel Stokes (1819–1903) was born in Ireland but for most of his life was at Cambridge University, first as a student and later as a professor. Stokes was one of the foremost applied mathemati- cians of the nineteenth century, best known for his work in fluid dynamics and the wave theory of light. The basic equations of fluid mechanics (the Navier–Stokes equations) are named partly in his honor, and one of the fundamental theorems of vector calculus bears his name. He was also one of the pioneers in the use of divergent (asymptotic) series. 8Robert A. Millikan (1868–1953) was educated at Oberlin College and Columbia University. Later he was a professor at the University of Chicago and California Institute of Technology. His determination of the charge on an electron was published in 1910. For this work, and for other studies of the photoelectric effect, he was awarded the Nobel Prize for Physics in 1923. August 7, 2012 21:03 c02 Sheet number 37 Page number 67 cyan black 2.3 Modeling with First Order Equations 67 (c) Suppose that the resistive force is k|v|, where v is measured in m/s and k is a constant. If the mass is dropped from a height of 30 m and must hit the ground with a velocity of no more than 10 m/s, determine the coefficient of resistance k that is required. 29. Suppose that a rocket is launched straight up from the surface of the earth with initial velocity v0 = √ 2gR, where R is the radius of the earth. Neglect air resistance. (a) Find an expression for the velocity v in terms of the distance x from the surface of the earth. (b) Find the time required for the rocket to go 240,000 mi (the approximate distance from the earth to the moon). Assume that R = 4000 mi. 30. Let v(t) and w(t) be the horizontal and vertical components, respectively, of the velocity of a batted (or thrown) baseball. In the absence of air resistance,v and w satisfy the equations dv/dt = 0, dw/dt = −g. (a) Show that v = u cos A, w = −gt + u sin A, where u is the initial speed of the ball and A is its initial angle of elevation. (b) Let x(t) and y(t) be the horizontal and vertical coordinates, respectively, of the ball at time t. If x(0) = 0 and y(0) = h, find x(t) and y(t) at any time t. (c) Let g = 32 ft/s2, u = 125 ft/s, and h = 3 ft. Plot the trajectory of the ball for several values of the angle A; that is, plot x(t) and y(t) parametrically. (d) Suppose the outfield wall is at a distance L and has height H . Find a relation between u and A that must be satisfied if the ball is to clear the wall. (e) Suppose that L = 350 ft and H = 10 ft. Using the relation in part (d), find (or estimate from a plot) the range of values of A that correspond to an initial velocity of u = 110 ft/s. (f) For L = 350 and H = 10, find the minimum initial velocity u and the corresponding optimal angle A for which the ball will clear the wall. 31. A more realistic model (than that in Problem 30) of a baseball in flight includes the effect of air resistance. In this case the equations of motion are dv/dt = −rv, dw/dt = −g −rw, where r is the coefficient of resistance. (a) Determine v(t) and w(t) in terms of initial speed u and initial angle of elevation A. (b) Find x(t) and y(t) if x(0) = 0 and y(0) = h. (c) Plot the trajectory of the ball for r = 1/5, u = 125, h = 3, and for several values of A. How do the trajectories differ from those in Problem 31 with r = 0? (d) Assuming that r = 1/5 and h = 3, find the minimum initial velocity u and the optimal angle A for which the ball will clear a wall that is 350 ft distant and 10 ft high. Compare this result with that in Problem 30(f). 32. Brachistochrone Problem. One of the famous problems in the history of mathematics is the brachistochrone9 problem: to find the curve along which a particle will slide without friction in the minimum time from one given point P to another Q, the second point being lower than the first but not directly beneath it (see Figure 2.3.6). This problem was posed by Johann Bernoulli in 1696 as a challenge problem to the mathematicians of his day. 9The word “brachistochrone” comes from the Greek words brachistos, meaning shortest, and chronos, meaning time. • • • August 7, 2012 21:03 c02 Sheet number 38 Page number 68 cyan black 68 Chapter 2. First Order Differential Equations Correct solutions were found by Johann Bernoulli and his brother Jakob Bernoulli and by Isaac Newton, Gottfried Leibniz, and the Marquis de L’Hôpital. The brachistochrone problem is important in the development of mathematics as one of the forerunners of the calculus of variations. In solving this problem, it is convenient to take the origin as the upper point P and to orient the axes as shown in Figure 2.3.6. The lower point Q has coordinates (x0, y0). It is then possible to show that the curve of minimum time is given by a function y = φ(x) that satisfies the differential equation (1 + y′ 2)y = k2, (i) where k2 is a certain positive constant to be determined later. (a) Solve Eq. (i) for y′. Why is it necessary to choose the positive square root? (b) Introduce the new variable t by the relation y = k2 sin2 t. (ii) Show that the equation found in part (a) then takes the form 2k2 sin2 t dt = dx. (iii) (c) Letting θ = 2t, show that the solution of Eq. (iii) for which x = 0 when y = 0 is given by x = k2(θ −sin θ)/2, y = k2(1 −cos θ)/2. (iv) Equations (iv) are parametric equations of the solution of Eq. (i) that passes through (0, 0). The graph of Eqs. (iv) is called a cycloid. (d) If we make a proper choice of the constant k, then the cycloid also passes through the point (x0, y0)and is the solution of the brachistochrone problem. Find k if x0 = 1 and y0 = 2. xP y Q(x0 , y0 ) FIGURE 2.3.6 The brachistochrone. 2.4 Differences Between Linear and Nonlinear Equations Up to now, we have been primarily concerned with showing that first order differ- ential equations can be used to investigate many different kinds of problems in the natural sciences, and with presenting methods of solving such equations if they are either linear or separable. Now it is time to turn our attention to some more generalmmmm