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Chapter 28: Problem 45

Suppose that, after falling for \(13 \mathrm{s}\), the parachutist from Examples 1.1 and 1.2 pulls the rip cord. At this point, assume that the drag coefficient is instantaneously increased to a constant value of \(55 \mathrm{kg} / \mathrm{s}\). Compute the parachutist's velocity from \(t=0\) to \(30 \mathrm{s}\) with Heun's method (without iteration of the corrector) using a step-size of 2 s. Plot \(v\) versus \(t\) for \(t=0\) to 30 s.

Short Answer

Expert verified
To compute the parachutist's velocity from t=0 to 30 seconds using Heun's method without iteration of the corrector and a step-size of 2 seconds, we need to apply Heun's method to the provided differential equation: \( \frac{dv}{dt} = g - \frac{c}{m}v \). Heun's method consists of the predictor step using Euler's method and the corrector step using the average slope. Apply these steps to calculate the velocity for each time point before and after the parachute opens, updating the drag coefficient at t=13 seconds. Finally, plot the velocity vs. time graph for t=0 to 30 seconds.

Step by step solution

01

Understand the problem and parameters

We have a given drag coefficient of 55 kg/s for the parachutist after the ripcord is pulled, and an initial velocity of 0 m/s. The parachute opens at 13 seconds, and we need to compute the velocity from t=0 to t=30 seconds. We apply Heun's method with a step-size of 2 seconds.
02

Set up the differential equation for the problem

The equation for the velocity of the parachutist is given by the differential equation: \( \frac{dv}{dt} = g - \frac{c}{m}v \) where \(g\) is the acceleration due to gravity (9.81 m/s虏), and \(c/m\) represents the drag coefficient of the parachutist that changes at t=13 s. Before the parachute opens, \(c/m = 0\), and after the parachute opens, \(c/m = 55/68\) kg/s (we have been given that 68 kg is the mass of the parachutist).
03

Apply the Heun's method to compute the velocity

The Heun's method without iteration of the corrector can be described as follows: 1. Calculate the predictor step using Euler's method: \(v_{p}(t+\Delta{t}) = v(t) + \Delta{t}*\frac{dv}{dt}(t)\) 2. Correct the predictor step using the average slope: \(v(t+\Delta{t}) = v(t) + \frac{\Delta{t}}{2}*(\frac{dv}{dt}(t) + \frac{dv}{dt}(t+\Delta{t}))\) Now we will apply Heun's method to compute the velocity for t=0 to 30 seconds with a step size of 2 seconds: For \(t=0\), before the parachute opens: 1. \(v(0) = 0\) 2. Find the value of \(\frac{dv}{dt}(0)\) using the differential equation 3. Calculate predictor step using Euler's method 4. Correct the predictor step using the average slope Repeat these steps for t=2, 4, ..., 12 seconds (before the parachute opens). After the parachute is opened at t=13 seconds, the drag coefficient changes. We will need to use the updated drag coefficient to calculate \(\frac{dv}{dt}(t)\) for t=14, 16, ..., 30 seconds.
04

Plot the velocity vs. time graph

After calculating the velocity at each time point, we can use a graphing tool to plot the velocity vs. time graph for t=0 to 30 seconds. The solution should consist of a step-by-step instruction on applying Heun's method to calculate the velocity at each time step, and a plot of the computed velocities for the given time range.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Equations
Differential equations are mathematical equations that relate a function with its derivatives. These equations are essential in modeling real-world phenomena where change rates are involved. In the context of the parachutist problem, the differential equation describes how the velocity of the parachutist changes over time as they fall. The equation is: \[ \frac{dv}{dt} = g - \frac{c}{m}v \] where \(v\) is the velocity, \(g\) is the gravitational acceleration, \(c\) is the drag coefficient, and \(m\) is the mass of the parachutist. Initially, when the parachutist begins to fall without a deployed parachute, resistance is minimal. However, upon deploying the parachute, the drag coefficient increases significantly, reducing the acceleration and, eventually, the velocity. Therefore, understanding differential equations allows us to predict how these changes occur over time.
Parachutist Problem
The parachutist problem involves calculating the velocity of a person falling through the air after jumping from a height. It is a classic example of modeling motion affected by gravitational and drag forces. Initially, the parachutist falls freely, allowing gravity to increase their speed. At a certain point, typically when reaching a certain velocity or after a specific time, the parachute is deployed. The deployment of the parachute increases air resistance, drastically changing the dynamics of the fall. This resistance is expressed in the form of a drag coefficient, which impacts the velocity calculation after the parachute is opened. Solving this problem often involves balancing these differing forces to compute how velocity changes over time, considering the varying drag.
Numerical Methods
Numerical methods are techniques used to approximate solutions for mathematical problems that may not have exact solutions. In the parachutist problem, Heun's method, a form of numerical technique, is used to solve the velocity equation. -Heun's method is an improvement of Euler's method and involves two main steps: * The **predictor step**, where an initial estimate is made. * The **corrector step**, where the estimate is improved based on the average rate of change. -These steps are repeated over the desired time range to provide a sequence of estimates that approximate the true solution. Numerical methods are vital in engineering and physics, helping compute challenging problems where analytical solutions are hard to obtain.
Velocity Calculation
In the parachutist problem, velocity calculation is essential for understanding how fast the parachutist is moving at any given time. The initial velocity starts at zero since the parachutist begins from rest. As time progresses and the parachute deploys, a greater drag force applies, slowing the parachutist's fall. Calculating the velocity involves applying Heun's method through repeated calculations: - **Step Size**: We use a step-size of 2 seconds for incremental calculations. - **Time Intervals**: Apply calculations at specific intervals (e.g., 0 to 2 seconds, 2 to 4 seconds) till the final time (30 seconds). - **Drag Changes**: Consider the altered drag coefficient after the parachute opens to adjust the slope calculations. By carefully executing these steps, we can find the velocity at each time point, ultimately allowing us to plot a curve showing how velocity evolves up to 30 seconds.

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Most popular questions from this chapter

In the investigation of a homicide or accidental death, it is often important to estimate the time of death. From the experimental observations, it is known that the surface temperature of an object changes at a rate proportional to the difference between the temperature of the object and that of the surrounding environment or ambient temperature. This is known as Newton's law of cooling. Thus, if \(T(t)\) is the temperature of the object at time \(t,\) and \(T_{a}\) is the constant ambient temperature: $$\frac{d T}{d t}=-K\left(T-T_{a}\right)$$ where \(K>0\) is a constant of proportionality. Suppose that at time \(t=0\) a corpse is discovered and its temperature is measured to be \(T_{o}\) We assume that at the time of death, the body temperature, \(T_{d}\) was at the normal value of \(37^{\circ} \mathrm{C}\). Suppose that the temperature of the corpse when it was discovered was \(29.5^{\circ} \mathrm{C}\), and that two hours later, it is \(23.5^{\circ} \mathrm{C}\). The ambient temperature is \(20^{\circ} \mathrm{C}\) (a) Determine \(K\) and the time of death. (b) Solve the ODE numerically and plot the results.

If \(c_{\mathrm{in}}=c_{b}\left(1-e^{-0.12 t}\right),\) calculate the outflow concentration of a conservative substance (no reaction) for a single, completely mixed reactor as a function of time. Use Heun's method (without iteration) to perform the computation. Employ values of \(c_{b}=40 \mathrm{mg} / \mathrm{m}^{3}\) \(Q=6 \mathrm{m}^{3} / \mathrm{min}, V=100 \mathrm{m}^{3},\) and \(c_{0}=20 \mathrm{mg} / \mathrm{m}^{3} .\) Perform the computation from \(t=0\) to 100 min using \(h=2 .\) Plot your results along with the inflow concentration versus time.

A forced damped spring-mass system (Fig. \(\mathrm{P} 28.47\) ) has the following ordinary differential equation of motion: \\[ m \frac{d^{2} x}{d t^{2}}+a\left|\frac{d x}{d t}\right| \frac{d x}{d t}+k x=F_{o} \sin (\omega t) \\] where \(x=\) displacement from the equilibrium position, \(t=\) time, \(m=2 \mathrm{kg}\) mass, \(a=5 \mathrm{N} /(\mathrm{m} / \mathrm{s})^{2},\) and \(k=6 \mathrm{N} / \mathrm{m} .\) The damping term is nonlinear and represents air damping. The forcing function \(F_{o} \sin (\omega t)\) has values of \(F_{o}=2.5 \mathrm{N}\) and \(\omega=0.5 \mathrm{rad} / \mathrm{sec} .\) The initial conditions are Initial velocity \(\frac{d x}{d t}=0 \mathrm{m} / \mathrm{s}\) Initial displacement \(x=1 \mathrm{m}\) Solve this equation using a numerical method over the time period \(0 \leq t \leq 15\) s. Plot the displacement and velocity versus time, and plot the forcing function on the same curve. Also, develop a separate plot of velocity versus displacement.

The Lotka-Volterra equations described in Sec. 28.2 have been refined to include additional factors that impact predator-prey dynamics. For example, over and above predation, prey population can be limited by other factors such as space. Space limitation can be incorporated into the model as a carrying capacity (recall the logistic model described in Prob. 28.16 ) as in \\[ \frac{d x}{d t}=a\left(1-\frac{x}{K}\right) x-b x y \\] \\[ \frac{d y}{d t}=-c y+d x y \\] where \(K=\) the carrying capacity. Use the same parameter values and initial conditions as in Sec. 28.2 to integrate these equations from \(t=0\) to 100 using ode 45 (a) Employ a very large value of \(K=10^{8}\) to validate that you obtain the same results as in Sec. 28.2 (b) Compare (a) with the more realistic carrying capacity of \(K=\) \(20,000 .\) Discuss your results.

Bacteria growing in a batch reactor utilize a soluble food source (substrate) as depicted in Fig. P28.16. The uptake of the substrate is represented by a logistic model with Michaelis-Menten limitation. Death of the bacteria produces detritus which is subsequently converted to the substrate by hydrolysis. In addition, the bacteria also excrete some substrate directly. Death, hydrolysis and excretion are all simulated as first-order reactions. Mass balances can be written as \(\frac{d X}{d t}=\mu_{\max }\left(1-\frac{X}{K}\right)\left(\frac{S}{K_{s}+S}\right) X-k_{d} X-k_{e} X\) \(\frac{d C}{d t}=k_{d} X-k_{h} C\) \(\frac{d S}{d t}=k_{e} X+k_{h} C-\mu_{\max }\left(1-\frac{X}{K}\right)\left(\frac{S}{K_{s}+S}\right) X\) where \(X, C,\) and \(S=\) the concentrations \([\mathrm{mg} / \mathrm{L}]\) of bacteria, detritus, and substrate, respectively; \(\mu_{\max }=\) maximum growth rate [/d], \(K=\) the logistic carrying capacity \([\mathrm{mg} / \mathrm{L}] ; K_{s}=\) the Michaelis-Menten half-saturation constant \([\mathrm{mg} / \mathrm{L}], k_{d}=\) death rate \([/ \mathrm{d}] ; k_{e}=\) excretion rate \([/ \mathrm{d}] ;\) and \(k_{h}=\) hydrolysis rate \([/ \mathrm{d}] .\) Simulate the concentrations from \(t=0\) to 100 d, given the initial conditions \(X(0)=1 \mathrm{mg} / \mathrm{L}, S(0)=\) \(100 \mathrm{mg} / \mathrm{L}\) and \(C(0)=0 \mathrm{mg} / \mathrm{L} .\) Employ the following parameters in your calculation: \(\mu_{\max }=10 / \mathrm{d}, K=10 \mathrm{mg} / \mathrm{L}, K_{s}=10 \mathrm{mg} / \mathrm{L}, k_{d}=\) \(0.1 / \mathrm{d}, k_{e}=0.1 / \mathrm{d},\) and \(k_{h}=0.1 / \mathrm{d}\).
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