Question

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

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

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.

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: \(640\,\text{lb} \cdot 0.453592\,\frac{\text{kg}}{\text{lb}} = 290.297 \,\text{kg}\)

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: \(F_{motor} - F_{resistance} = ma\)

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): \(F_{motor} - 1.5v_{\text{N/s}} = ma\) \(a = \frac{F_{motor} - 1.5v_{\text{N/s}}}{m}\)

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: \(F_{motor} = 20\,\text{lb} \cdot 4.44822\,\frac{\text{N}}{\text{lb}} = 89 \,\text{N}\)

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.

Key concepts

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.