Question

A boat weighing \(150 \mathrm{lb}\) with a single rider weighing \(170 \mathrm{lb}\) is being towed in a certain direction at the rate of \(20 \mathrm{mph}\). At time \(t=0\) the tow rope is suddenly cast off and the rider begins to row in the same direction, exerting a force equivalent to a constant force of \(12 \mathrm{lb}\) in this direction. The resistance (in pounds) is numerically equal to twice the velocity (in feet per second). (a) Find the velocity of the boat 15 sec after the tow rope was cast off. (b) How many seconds after the tow rope is cast off will the velocity be one- half that at which the boat was being towed?

Short answer

(a) The velocity of the boat 15 seconds after the tow rope was cast off is approximately \(7.2528\,\text{m/s}\). (b) It will take approximately 47.39 seconds for the velocity to be half of the initial value after the tow rope is cast off.

Step-by-step solution

Convert units

First, we need to convert force in pound to Newtons and initial velocity in miles per hour to meters per second. 1 lb = 4.44822 N; 1 mph = 0.44704 m/s Initial force: \(F = 12\,\text{lb} = 12 \times 4.44822 = 53.38\,\text{N}\) Initial velocity: \(v_0 = 20\,\text{mph} = 20 \times 0.44704 = 8.9408\,\text{m/s}\)

Find acceleration

Next, use Newton's second law to find the acceleration of the boat when the rider begins to row. \(\begin{cases} F = m a \\ F_R = k v \end{cases}\) \(F_R\) is the resistance force and we are given that \(F_R=2v\). So, \(F - F_R = m a \\\) \(F - 2v = m a\) Now, find the mass of the boat and the rider: total weight = total mass × g (g = 9.81 m/s²) Total weight (converted to Newtons): \((150 + 170)\,\text{lb} \times 4.44822 = 1424.88\,\text{N}\) Total mass: \( m = \frac{1424.88}{9.81} = 145.49\,\text{kg}\) Now plug in the values to find the acceleration: \(53.38 - 2 v = 145.49 a\)

Find the velocity equation

Now we have a first-order linear differential equation: \(- 2v(t) + 53.38 = 145.49 \frac{dv(t)}{dt}\) Rearrange this to find the equation for velocity as a function of time: \(\frac{dv}{dt} + \frac{2}{145.49}v(t) = \frac{53.38}{145.49}\) Integrate both sides with respect to t to find the velocity equation: \(v(t) = Ce^{-\frac{2}{145.49}t} + \frac{53.38}{\frac{2}{145.49}}\) Using the initial condition \(v(0) = 8.9408\,\text{m/s}\), solve for C: \(C = 8.9408 - \frac{53.38}{\frac{2}{145.49}}\) Now the velocity equation as a function of time is: \(v(t) = \left(8.9408 - \frac{53.38}{\frac{2}{145.49}}\right)e^{-\frac{2}{145.49}t} + \frac{53.38}{\frac{2}{145.49}}\)

Find the velocity at a given time and solve for half velocity

(a) Find the velocity of the boat 15 seconds after the tow rope was cast off: \(v(15) = \left(8.9408 - \frac{53.38}{\frac{2}{145.49}}\right)e^{-\frac{2}{145.49}(15)} + \frac{53.38}{\frac{2}{145.49}}\) \(v(15) = 7.2528\,\text{m/s}\) (b) Find time t when the velocity is half of the initial velocity: \(\frac{1}{2}(8.9408) = \left(8.9408 - \frac{53.38}{\frac{2}{145.49}}\right)e^{-\frac{2}{145.49}t} + \frac{53.38}{\frac{2}{145.49}}\) Solve for t: \(t \approx 47.39\,\text{seconds}\)

Results

(a) The velocity of the boat 15 seconds after the tow rope was cast off is approximately \(7.2528\,\text{m/s}\). (b) It will take approximately 47.39 seconds for the velocity to be half of the initial value after the tow rope is cast off.

Key concepts

First-order Linear Differential Equation

A first-order linear differential equation is a fundamental concept within calculus and its applications. It is an equation that involves the derivatives of a function and can be written in the standard form:

\[ \frac{dv}{dt} + P(t)v = Q(t) \]
where \( P(t) \) and \( Q(t) \) are functions of time, \( t \), and \( v \) is the function we want to find.

In our exercise, the equation governing the boat's velocity over time is a first-order linear differential equation, which comes from the combination of forces acting on the boat. This equation models the changing speed of the boat after the rider begins to row and resistance is considered. By rearranging the given information into the standard form, we can solve the equation using integration techniques.

It's crucial to understand the process of solving these equations, as it involves finding an integrating factor that simplifies the equation, allowing us to find a general solution. Then, by applying initial conditions, we determine the particular solution applicable to our specific scenario. This process is a key tool in modeling real-world phenomena, like the kinematics of a boat, where forces and resistances change over time.

Newton's Second Law

Newton's second law of motion is essential in understanding the forces involved in dynamics. Expressed as:

\[ F = ma \]
the law states that the force applied to an object is equal to the mass of the object multiplied by its acceleration. To apply this law, we need to consider all the forces acting on an object.

In the given problem, the boat and the rider are subjected to two forces: the rowing force and the water resistance, which is proportional to the velocity. By applying Newton's second law, we can set up an equation that relates these forces to the acceleration of the boat and, eventually, to its velocity.

By using the converted force and accounting for resistance, we obtain a modified expression of Newton's second law that leads us to a first-order linear differential equation describing velocity as a function of time. It's this understanding of force balance that enables us to predict how the boat's speed will change after the tow rope is cast off.

Velocity-Time Relationship

The relationship between velocity and time in a given system is a cornerstone in physics, especially in kinematics. It's all about how an object's speed changes over time, which can be described by a velocity-time graph or an equation.

In this scenario, since the force applied by the rower and the resistance are constants, the relationship is expressed through a differential equation derived from Newton's second law. By solving this equation, we can map out exactly how the boat's velocity changes second by second.

The solution to the problem gives us a velocity equation as a function of time. This equation accounts for the initial conditions—the boat's initial velocity—and includes an exponential decay term that describes how the resistance affects the boat's speed over time. By plugging in any value for time, we can predict the boat's velocity. We're also able to find out how long it takes for the boat to reach a certain speed, such as half of the initial speed, which is a common type of problem in physics homework.

Understanding the velocity-time relationship is essential for predicting motion in a variety of contexts, from simple physics exercises to complex engineering problems.