You are working as a crime scene investigator and must predict the temperature of a

Use calculus to verify that Eq. (1.9) is a solution of Eq. (1.8) for the initial condition v(0) 0.

Use calculus to solve Eq. (1.21) for the case where the initial velocity is (a) positive and (b) negative. (c) Based on your results for (a) and (b), perform the same computation as in Example 1.1 but with an initial velocity of 40 m/s. Compute values of the velocity from t 0 to 12 s at intervals of 2 s. Note that for this case, the zero velocity occurs at t 3.470239 s. 1

The following information is available for a bank account: Date Deposits Withdrawals Balance 5/1 1512.33 220.13 327.26 6/1 216.80 378.61 7/1 450.25 106.80 8/1 127.31 350.61 9/1 Note that the money earns interest which is computed as Interest = i Bi where i the interest rate expressed as a fraction per month, and Bi the initial balance at the beginning of the month. (a) Use the conservation of cash to compute the balance on 6/1, 7/1, 8/1, and 9/1 if the interest rate is 1% per month (i 0.01/month). Show each step in the computation. (b) Write a differential equation for the cash balance in the form d B dt = f [D(t), W(t),i] where t time (months), D(t) deposits as a function of time ($/month), W(t) withdrawals as a function of time ($/month). For this case, assume that interest is compounded continuously; that is, interest iB. (c) Use Eulers method with a time step of 0.5 month to simulate the balance. Assume that the deposits and withdrawals are applied uniformly over the month. (d) Develop a plot of balance versus time for (a) and (c). 1.4

Repeat Example 1.2. Compute the velocity to t = 12 s, with a step size of (a) 1 and (b) 0.5 s. Can you make any statement regarding the errors of the calculation based on the results?

Rather than the nonlinear relationship of Eq. (1.7), you might choose to model the upward force on the bungee jumper as a linear relationship: FU = c v where c = a first-order drag coefficient (kg/s). (a) Using calculus, obtain the closed-form solution for the case where the jumper is initially at rest (v = 0 at t = 0). (b) Repeat the numerical calculation in Example 1.2 with the same initial condition and parameter values. Use a value of 11.5 kg/s for c

For the free-falling bungee jumper with linear drag (Prob. 1.5), assume a first jumper is 70 kg and has a drag coefficient of 12 kg/s. If a second jumper has a drag coefficient of 15 kg/s and a mass of 80 kg, how long will it take her to reach the same velocity jumper 1 reached in 9 s?

For the second-order drag model (Eq. 1.8), compute the velocity of a free-falling parachutist using Eulers method for the case where m = 80 kg and cd = 0.25 kg/m. Perform the calculation from t = 0 to 20 s with a step size of 1 s. Use an initial condition that the parachutist has an upward velocity of 20 m/s at t = 0. At t = 10 s, assume that the chute is instantaneously deployed so that the drag coefficient jumps to 1.5 kg/m.

The amount of a uniformly distributed radioactive contaminant contained in a closed reactor is measured by its concentration c (becquerel/liter or Bq/L). The contaminant decreases at a decay rate proportional to its concentration; that is Decay rate = kc where k is a constant with units of day1 . Therefore, according to Eq. (1.14), a mass balance for the reactor can be written as dc dt = kc change in mass = decrease by decay (a) Use Eulers method to solve this equation from t = 0 to 1 d with k = 0.175 d1. Employ a step size of t = 0.1 d. The concentration at t = 0 is 100 Bq/L. (b) Plot the solution on a semilog graph (i.e., ln c versus t) and determine the slope. Interpret your results. 1.9

A storage tank (Fig. P1.9) contains a liquid at depth y where y = 0 when the tank is half full. Liquid is withdrawn at a constant flow rate Q to meet demands. The contents are resupplied at a sinusoidal rate 3Q sin2 (t). Equation (1.14) can be written for this system as d(Ay) dt = 3Q sin2(t) Q change in volume = (in o w) (out o w) or, since the surface area A is constant dy dt = 3 Q A sin2(t) Q A Use Eulers method to solve for the depth y from t = 0 to 10 d with a step size of 0.5 d. The parameter values are A = 1250 m2 and Q = 450 m3 /d. Assume that the initial condition is y = 0.

Apply the conservation of volume (see Prob. 1.9) to simulate the level of liquid in a conical storage tank (Fig. P1.11). The liquid flows in at a sinusoidal rate of Qin = 3 sin2 (t) and flows out according to Qout = 3(y yout) 1.5 y > yout Qout = 0 y yout where flow has units of m3 /d and y = the elevation of the water surface above the bottom of the tank (m). Use Eulers method to solve for the depth y from t = 0 to 10 d with a step size of 0.5 d. The parameter values are rtop 2.5 m, ytop 4 m, and yout 1 m. Assume that the level is initially below the outlet pipe with y(0) 0.8 m. 1.1

A group of 35 students attend a class in an insulated room which measures 11 m by 8 m by 3 m. Each student takes up about 0.075 m3 and gives out about 80 W of heat (1 W = 1 J/s). Calculate the air temperature rise during the first 20 minutes of the class if the room is completely sealed and insulated. Assume the heat capacity Cv for air is 0.718 kJ/(kg K). Assume air is an ideal gas at 20C and 101.325 kPa. Note that the heat absorbed by the air Q is related to the mass of the air m the heat capacity, and the change in temperature by the following relationship: Q = m T2 T1 CvdT = mCv(T2 T1) The mass of air can be obtained from the ideal gas law: PV = m Mwt RT where P is the gas pressure, V is the volume of the gas, Mwt is the molecular weight of the gas (for air, 28.97 kg/kmol), and R is the ideal gas constant [8.314 kPa m3 /(kmol K)].

Figure P1.13 depicts the various ways in which an average man gains and loses water in one day. One liter is ingested as food, and the body metabolically produces 0.3 liters. In breathing air, the exchange is 0.05 liters while inhaling, and 0.4 liters while exhaling over a one-day period. The body will also lose 0.3, 1.4, 0.2, and 0.35 liters through sweat, urine, feces, and through the skin, respectively. To maintain steady state, how much water must be drunk per day?

In our example of the free-falling bungee jumper, we assumed that the acceleration due to gravity was a constant value of 9.81 m/s2 . Although this is a decent approximation when we are examining falling objects near the surface of the earth, the gravitational force decreases as we move above sea level. A more general representation based on Newtons inverse square law of gravitational attraction can be written as g(x) = g(0) R2 (R + x)2 where g(x) = gravitational acceleration at altitude x (in m) measured upward from the earths surface (m/s2 ), g(0) = gravitational acceleration at the earths surface (= 9.81 m/s2 ), and R = the earths radius (= 6.37 106 m). (a) In a fashion similar to the derivation of Eq. (1.8), use a force balance to derive a differential equation for velocity as a function of time that utilizes this more complete representation of gravitation. However, for this derivation, assume that upward velocity is positive. (b) For the case where drag is negligible, use the chain rule to express the differential equation as a function of altitude rather than time. Recall that the chain rule is dv dt = dv dx dx dt (c) Use calculus to obtain the closed form solution where v = v0 at x = 0. (d) Use Eulers method to obtain a numerical solution from x = 0 to 100,000 m using a step of 10,000 m where the initial velocity is 1500 m/s upward. Compare your result with the analytical solution.

Suppose that a spherical droplet of liquid evaporates at a rate that is proportional to its surface area. dV dt = k A where V = volume (mm3 ), t = time (min), k = the evaporation rate (mm/min), and A = surface area (mm2 ). Use Eulers method to compute the volume of the droplet from t = 0 to 10 min using a step size of 0.25 min. Assume that k = 0.08 mm/min and that the droplet initially has a radius of 2.5 mm. Assess the validity of your results by determining the radius of your final computed volume and verifying that it is consistent with the evaporation rate.

A fluid is pumped into the network shown in Fig. P1.16. If Q2 = 0.7, Q3 = 0.5, Q7 = 0.1, and Q8 = 0.3 m3 /s, determine the other flows.

Newtons law of cooling says that the temperature of a body changes at a rate proportional to the difference between its temperature and that of the surrounding medium (the ambient temperature), dT dt = k(T Ta) where T = the temperature of the body (C), t = time (min), k = the proportionality constant (per minute), and Ta = the ambient temperature (C). Suppose that a cup of coffee originally has a temperature of 70 C. Use Eulers method to compute the temperature from t = 0 to 20 min using a step size of 2 min if Ta = 20 C and k = 0.019/min.

You are working as a crime scene investigator and must predict the temperature of a homicide victim over a 5-hour period. You know that the room where the victim was found was at 10 C when the body was discovered. (a) Use Newtons law of cooling (Prob. 1.17) and Eulers method to compute the victims body temperature for the 5-hr period using values of k 0.12/hr and t 0.5 hr. Assume that the victims body temperature at the time of death was 37 C, and that the room temperature was at a constant value of 10 C over the 5-hr period. (b) Further investigation reveals that the room temperature had actually dropped linearly from 20 to 10 C over the 5-hr period. Repeat the same calculation as in (a) but incorporate this new information. (c) Compare the results from (a) and (b) by plotting them on the same graph.

The velocity is equal to the rate of change of distance, x (m): dx dt = v(t) (P1.19) Use Eulers method to numerically integrate Eq. (P1.19) and (1.8) in order to determine both the velocity and distance fallen as a function of time for the first 10 seconds of freefall using the same parameters and conditions as in Example 1.2. Develop a plot of your results.