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Chapter 9: Problem 17

A sailboat is running along a straight course with the wind providing a constant forward force of \(200 \mathrm{N}\). The only other force acting on the boat is resistance as the boat moves through the water. The resisting force is numerically equal to fifty times the boat's speed, and the initial velocity is \(1 \mathrm{m} / \mathrm{s}\). What is the maximum velocity in meters per second of the boat under this wind?

Short Answer

Expert verified
The maximum velocity of the sailboat is 4 m/s.

Step by step solution

01

Define the Problem

We need to find the maximum velocity of the sailboat given a constant driving force from the wind and a resistance force dependent on the speed of the boat. The driving force is 200 N, and the resistance force is given by 50 times the boat's speed.
02

Set Up the Force Balance Equation

At maximum velocity, the net force acting on the boat is zero. This is because the driving force and the resistance force are equal. Thus, we write:\[ \text{Driving Force} = \text{Resistance Force} \]\[ 200 = 50v \]where \(v\) is the velocity of the boat.
03

Solve for Maximum Velocity

We solve the equation from Step 2:\[ 200 = 50v \]Divide both sides by 50:\[ v = \frac{200}{50} \]\[ v = 4 \]
04

Conclusion

The sailboat reaches its maximum velocity when the resistance force equals the driving force, resulting in a velocity where net force is zero.

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Key Concepts

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

Forces in Motion
Understanding forces in motion is crucial when considering how objects like sailboats move through their environment. In the case of the sailboat, the primary force propelling it forward is from the wind. This force is constant and is measured to be 200 N. Simultaneously, as the sailboat moves, it experiences a resisting force emanating from the water. This resistance opposes the boat's motion by creating drag. To comprehend motion fully, remember that:
  • Forces can be thought of as pushes or pulls upon an object resulting from interaction with other objects.
  • The direction and strength of these forces determine how the object's motion changes.
  • Analyzing forces helps us predict the motion of objects under various conditions.
Learning about how forces act on moving objects is fundamental in understanding the dynamics of motion.
Maximum Velocity
The concept of maximum velocity is tied directly to the idea of balanced forces. For a sailboat on a wind-powered journey, the maximum velocity is reached when the net force on the boat is zero. At this point, the driving force from the wind equals the resisting force due to water drag.

To achieve maximum velocity:
  • A constant push (or driving force) is countered perfectly by an opposing force (resistance or drag).
  • When these forces balance each other out, the object stops accelerating, maintaining a constant speed.
  • The formula to find maximum velocity in this situation is given by the driving force divided by the factor determining resistance per unit speed.
In our example:
  • The driving force = 200 N.
  • The resistance = 50 times the speed (50v), leading us to maximum speed solving from: 200 = 50v.
  • Thus, v or maximum velocity = 4 m/s.
Maximum velocity is the plateau of a boat's speed only reached when resistant forces fully counter the push forward.
Resistance and Drag
Resistance and drag are commonplace in physics problems related to motion through fluids, whether air or water. For the sailboat, water provides drag, a resisting force which tries to slow the boat down.

Important aspects of resistance and drag include:
  • Resistance increases with speed. Hence, faster movement usually means more drag.
  • Drag force acts in the opposite direction to the object's motion, trying to reduce kinetic energy and slow it down.
  • The equation given in this problem, 50v, suggests drag is proportional to velocity. As speed goes up, so does the resistance.
Understanding these forces is crucial for solving problems that involve movement through different mediums, like air or water. This allows for predicting how fast a sailboat can go with a particular amount of force from the wind versus water resistance.
Equilibrium of Forces
Achieving equilibrium of forces means that all forces acting on an object are perfectly balanced, leading to a stasis in acceleration. When considering our sailboat problem, equilibrium is achieved at the point of maximum velocity.

Key points about equilibrium:
  • In equilibrium, the sum of all forces equals zero. It results in constant velocity or a rest state.
  • For our sailboat, equilibrium is when the forward force (200 N) equals the resisting force (50v).
  • This balance is crucial because it not only sets the boat's speed but also indicates stability in motion.
Understanding force equilibrium allows us to simplify complex dynamics into more manageable equations. When forces are balanced, objects maintain their current state of motion, whether that's at rest or moving at constant speed.

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Most popular questions from this chapter

Use a CAS to explore graphically each of the differential equations.Perform the following steps to help with your explorations. a. Plot a slope field for the differential equation in the given \(x y\) -window. b. Find the general solution of the differential equation using your CAS DE solver. c. Graph the solutions for the values of the arbitrary constant \(C=-2,-1,0,1,2\) superimposed on your slope field plot. d. Find and graph the solution that satisfies the specified initial condition over the interval \([0, b].\) e. Find the Euler numerical approximation to the solution of the initial value problem with 4 subintervals of the \(x\) -interval and plot the Euler approximation superimposed on the graph produced in part (d). f. Repeat part (e) for \(8,16,\) and 32 subintervals. Plot these three Euler approximations superimposed on the graph from part (e). g. Find the error ( \(y\) (exact) \(-y\) (Euler)) at the specified point \(x=b\) for each of your four Euler approximations. Discuss the improvement in the percentage error.$$\begin{aligned}&y^{\prime}=x+y, \quad y(0)=-7 / 10 ; \quad-4 \leq x \leq 4, \quad-4 \leq y \leq 4\\\&b=1\end{aligned}.$$

The fish and game department in a certain state is planning to issue hunting permits to control the deer population (one deer per permit). It is known that if the deer population falls below a certain level \(m,\) the deer will become extinct. It is also known that if the deer population rises above the carrying capacity \(M,\) the population will decrease back to \(M\) through disease and malnutrition. a. Discuss the reasonableness of the following model for the growth rate of the deer population as a function of time: $$\frac{d P}{d t}=r P(M-P)(P-m),$$ where \(P\) is the population of the deer and \(r\) is a positive constant of proportionality. Include a phase line. b. Explain how this model differs from the logistic model \(d P / d t=r P(M-P) .\) Is it better or worse than the logistic model? c. Show that if \(P > M\) for all \(t,\) then \(\lim _{t \rightarrow \infty} P(t)=M\). d. What happens if \(P < m\) for all \(t ?\) e. Discuss the solutions to the differential equation. What are the equilibrium points of the model? Explain the dependence of the steady-state value of \(P\) on the initial values of \(P .\) About how many permits should be issued?

Develop a model for the growth of trout and bass, assuming that in isolation trout demonstrate exponential decay [so that \(a<0\) in Equations (1a) and (1b)] and that the bass population grows logistically with a population limit \(M\). Analyze graphically the motion in the vicinity of the rest points in your model. Is coexistence possible?

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