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August 7, 2012 21:03 c02 Sheet number 30 Page number 60 cyan black

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?

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August 7, 2012 21:03 c02 Sheet number 31 Page number 61 cyan black

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.