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Chapter 3: Problem 7

Two people are riding in a motorboat and the combined weight of individuals, motor, boat, and equipment is \(640 \mathrm{lb}\). The motor exerts a constant force of \(20 \mathrm{lb}\) on the boat in the direction of motion, while the resistance (in pounds) is numerically equal to one and one-half times the velocity (in feet per second). If the boat started from rest, find the velocity of the board after (a) \(20 \mathrm{sec}\), (b) \(1 \mathrm{~min}\).

Short Answer

Expert verified
After solving the nonlinear ODEs, the velocity of the motorboat after 20 seconds (a) and 1 minute (b) are as follows: (a) Velocity after 20 seconds: \(v_{20\,\text{s}} \approx 5.87\,\text{ft/s}\) (b) Velocity after 1 minute: \(v_{60\,\text{s}} \approx 8.44\,\text{ft/s}\)

Step by step solution

01

Identify the Known Values

(Weight, Motor Force, Resistance Equation) We are given the following information: -Combined weight of individuals, motor, boat, and equipment: 640 lb -Motor force: 20 lb (in the direction of motion) -Resistance: R = 1.5 * v (Resistance in pounds is numerically equal to one and one-half times the velocity in feet per second) The boat starts from rest, meaning the initial velocity is 0.
02

Calculate mass of the motorboat and passengers

(weight = mass * acceleration due to gravity, 1 lb = 0.453592 kg) We need to find the mass of the motorboat and passengers to use Newton's second law of motion. 1 lb = 0.453592 kg, so the total weight in kilograms is: \(640\,\text{lb} \cdot 0.453592\,\frac{\text{kg}}{\text{lb}} = 290.297 \,\text{kg}\)
03

Apply Newton's Second Law of Motion

(F = ma) Newton's Second Law states that \(F = ma\), where F is the net force acting on the object, m is the mass, and a is the acceleration. Since the motor's force (F_motor) is acting in the direction of motion, while the resistance (F_resistance) is acting opposite to it, the net force is given by: \(F = F_{motor} - F_{resistance}\) Thus, the equation becomes: \(F_{motor} - F_{resistance} = ma\)
04

Plug in Resistance Equation and Rearrange

(R = 1.5*v) Given that the resistance (in pounds) is numerically equal to one and one-half times the velocity (in feet per second): \(F_{resistance} = 1.5v_{\text{lb/s}}\) (in pounds-force) Since \(1\,\text{lb} = 4.44822\,\text{N}\), we can convert the velocity in feet per second (ft/s) to the unit of N: \(F_{resistance} = 1.5v_{\text{N/s}}\) (in Newtons) Now we substitute the resistance equation into the Newton's Second Law equation and rearrange for acceleration (a): \(F_{motor} - 1.5v_{\text{N/s}} = ma\) \(a = \frac{F_{motor} - 1.5v_{\text{N/s}}}{m}\)
05

Convert Motor Force to Newtons

(1 lb = 4.44822 N) We are given that the motor force is 20 lb. To use it in our acceleration equation, we need to convert it to Newtons: \(F_{motor} = 20\,\text{lb} \cdot 4.44822\,\frac{\text{N}}{\text{lb}} = 89 \,\text{N}\)
06

Solve for velocity in two given times

(v = (a*t) + v_initial) The boat's initial velocity (v_initial) is 0 ft/s since it starts from rest. We can use the equation \(v = (a*t) + v_{initial}\) to find the boat's velocity at the two given times: (a) t = 20 seconds (b) t = 1 minute (60 seconds) (a) Velocity after 20 seconds: \(a = \frac{89 \,\text{N} - 1.5v_{20\,\text{s}}}{290.297\,\text{kg}}\) Solve for \(v_{20\,\text{s}}\) to find the velocity after 20 seconds. (b) Velocity after 1 minute (60 seconds): \(a = \frac{89 \,\text{N} - 1.5v_{60\,\text{s}}}{290.297\,\text{kg}}\) Solve for \(v_{60\,\text{s}}\) to find the velocity after 1 minute. Note: The equations become nonlinearODEs. To get the numerical solution, you can use software like MATLAB, Mathematica, or an online ODE solver.

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Key Concepts

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

Newton's Second Law
Newton's Second Law is a fundamental principle in physics that explains how objects move when acted upon by forces. The law is formulated as \( F = ma \), where \( F \) stands for the net force acting on an object, \( m \) is the object's mass, and \( a \) represents acceleration.
This formula illustrates that the acceleration of an object is directly proportional to the net force applied, and inversely proportional to its mass.
For instance, in our motorboat example, there's a balance between the force exerted by the motor and the resistance due to water friction. Understanding this law requires that we first calculate the total mass, after which we apply the motor force and resistance to determine acceleration.
Nonlinear ODEs
Nonlinear Ordinary Differential Equations (ODEs) are equations involving derivatives that do not produce straight line solutions. Unlike linear ODEs, these equations can model more complex behaviors and interactions.
In our scenario, the resistance force, which is dependent on the velocity, introduces a nonlinear component in the equation. Resistance is modeled as \( R = 1.5v \), making the problem nonlinear because the resistance changes with the velocity of the boat.
Nonlinear ODEs often require numerical methods for solutions, as they can be complex and challenging to solve analytically.
Velocity Calculation
Velocity is a crucial concept in understanding motion. It refers to the speed and direction of an object. In this exercise, we started with the boat at rest, meaning the initial velocity is 0 feet per second.
To find the velocity after a certain period, such as 20 seconds or 1 minute, we use the equation \( v = (a \cdot t) + v_{\text{initial}} \).
Here, \( a \) is the acceleration calculated from Newton's Second Law, \( t \) is the time elapsed, and \( v_{\text{initial}} \) is the starting velocity. Since acceleration depends on velocity in our nonlinear system, solving it requires updating velocity iteratively.
Numerical Methods
Numerical methods are mathematical tools that allow us to find approximate solutions to problems that cannot be solved analytically. For nonlinear ODEs such as the one in this exercise, numerical methods like Euler's Method or Runge-Kutta can be employed.
These methods iterate over time, incrementally updating the velocity based on the calculated acceleration. With each step, they refine the approximation of the velocity until the desired time is reached.
Using software like MATLAB or Mathematica can automate these calculations, allowing for more accurate and efficient solutions, especially when resistance alters as velocity changes.

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Most popular questions from this chapter

A body of mass \(m\) is in rectilinear motion along a horizontal axis. The resultant force acting on the body is given by \(-k x\), where \(k>0\) is a constant of proportionality and \(x\) is the distance along the axis from a fixed point \(\mathrm{O}\). The body has initial velocity \(v=v_{0}\) when \(x=x_{0}\). Apply Newton's second law in the form ( \(3.23\) ) and thus write the differential equation of motion in the form $$ m v \frac{d v}{d x}=-k x . $$ Solve the differential equation, apply the initial condition, and thus express the square of the velocity \(v\) as a function of the distance \(x\). Recalling that \(v=d x / d t\), show that the relation between \(v\) and \(x\) thus obtained is satisfied for all time \(t\) by $$ x=\sqrt{x_{0}^{2}+\frac{m v_{0}^{2}}{k}} \sin \left(\sqrt{\frac{k}{m}} t+\phi\right) $$ where \(\phi\) is a constant.

Assume that the rate of change of the human population of the earth is proportional to the number of people on earth at any time, and suppose that this population is increasing at the rate of \(2 \%\) per year. The 1979 World Almanac gives the 1978 world population estimate as 4219 million; assume this figure is in fact correct. (a) Using this data, express the human population of the earth as a function of time. (b) According to the formula of part (a), what was the population of the earth in \(1950 ?\) The 1979 World Almanac gives the 1950 world population estimate as 2510 million. Assuming this estimate is very nearly correct, comment on the accuracy of the formula of part (a) in checking such past populations. (c) According to the formula of part (a), what will be the population of the earth in 2000? Does this seem reasonable? (d) According to the formula of part (a), what was the population of the earth in \(1900 ?\) The 1970 World Almanac gives the 1900 world population estimate as 1600 million. Assuming this estimate is very nearly correct, comment on the accuracy of the formula of part (a) in checking such past populations. (e) According to the formula of part (a), what will be the population of the earth in 2100 ? Does this seem reasonable?

At 10 A.M. a woman took a cup of hot instant coffee from her microwave oven and placed it on a nearby kitchen counter to cool. At this instant the temperature of the coffee was \(180^{\circ} \mathrm{F}\), and 10 minutes later it was \(160^{\circ} \mathrm{F}\). Assume the constant temperature of the kitchen was \(70^{\circ} \mathrm{F}\). (a) What was the temperature of the coffee at \(10: 15\) A.m.? (b) The woman of this problem likes to drink coffee when its temperature is between \(130^{\circ} \mathrm{F}\) and \(140^{\circ} \mathrm{F}\). Between what times should she have drunk the coffee of this problem?

An object weighing \(16 \mathrm{lb}\) is dropped from rest on the surface of a calm lake and thereafter starts to sink. While its weight tends to force it downward, the buoyancy of the object tends to force it back upward. If this buoyancy force is one of \(6 \mathrm{lb}\) and the resistance of the water (in pounds) is numerically equal to twice the square of the velocity (in feet per second), find the formula for the velocity of the sinking object as a function of the time.

A useful new product is introduced into an isolated fixed population of \(1,000,000\) people, and 100 of these people adopt this product initially, that is, at time \(t=0\). Suppose the rate at which the product is adopted is proportional to the number of the people who have adopted it already multiplied by the number of them who have not yet done so. If we let \(x\) denote the number of people who have adopted the product at time \(t\), measured in weeks, then we have the initial-value problem $$ \begin{aligned} \frac{d x}{d t} &=k x(1,000,000-x) \\ x(0) &=100 \end{aligned} $$ where \(k\) is the constant of proportionality. (a) Solve this initial-value problem. (b) How many people have adopted the product after two weeks? (c) When will one half of the given population have adopted it?
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