A seaplane of total mass m lands on a lake with initial...

A seaplane of total mass $m$ lands on a lake with initial speed $v_{i}$ i. The only horizontal force on it is a resistive force on its pontoons from the water. The resistive force is proportional to the velocity of the seaplane: $\overline{\mathbf{R}}=-b \overrightarrow{\mathbf{v}}$ . Newton's second law applied to the plane is $-b v \hat{\mathbf{i}}=m(d v / d t) \hat{\mathbf{i}}$ . From the fundamental theorem of calculus, this differential equation implies that the speed changes according to $$\int_{v_{i}}^{v} \frac{d v}{v}=-\frac{b}{m} \int_{0}^{t} d t$$ (a) Carry out the integration to determine the speed of the seaplane as a function of time. (b) Sketch a graph of the speed as a function of time. (c) Does the seaplane come to a complete stop after a finite interval of time? (d) Does the seaplane travel a finite distance in stopping?

A seaplane of total mass $m$ lands on a lake with initial speed $v_{i}$ i. The only horizontal force on it is a resistive force on its pontoons from the water. The resistive force is proportional to the velocity of the seaplane: $\overline{\mathbf{R}}=-b \overrightarrow{\mathbf{v}}$ . Newton's second law applied to the plane is $-b v \hat{\mathbf{i}}=m(d v / d t) \hat{\mathbf{i}}$ . From the fundamental theorem of calculus, this differential equation implies that the speed changes according to $$\int_{v_{i}}^{v} \frac{d v}{v}=-\frac{b}{m} \int_{0}^{t} d t$$
(a) Carry out the integration to determine the speed of the seaplane as a function of time. (b) Sketch a graph of the speed as a function of time. (c) Does the seaplane come to a complete stop after a finite interval of time? (d) Does the seaplane travel a finite distance in stopping?

Key Concepts

Displacement via Integration

Determining the total distance traveled involves integrating the velocity function over time. Even though the velocity decays asymptotically, the cumulative displacement converges to a finite value, illustrating how integration can yield finite results from infinite time intervals.

Asymptotic Behavior

The concept of asymptotic behavior describes how a function, such as velocity in this case, approaches a limit—in this problem, zero—as time approaches infinity, without ever truly reaching that value in a finite amount of time.

Exponential Decay

Exponential decay arises naturally when a physical quantity decreases at a rate proportional to its current value. In this context, the integration of the differential equation yields an exponential decay function that describes how velocity decreases over time.

Resistive Force Proportional to Velocity

A resistive force that is proportional to the velocity represents a type of linear drag, where the magnitude of the force opposes the motion and increases with speed. This concept is common in modeling phenomena like air or water resistance under conditions of low Reynolds numbers.

Newton's Second Law

Newton's second law relates the net force acting on an object with its mass and the acceleration it experiences. In problems involving resistive forces, this law forms the foundation of the differential equation describing how the object's velocity changes over time.

Differential Equations and Integration

The problem leads to a first-order linear differential equation that describes the rate of change of velocity as a function of velocity itself. Solving such equations involves separating variables and integrating, which is a fundamental method in analyzing dynamic systems.

Transcript

00:01 And this is a rather ginormous problem, so let's get going.

00:02 It's problem of 643.

00:05 Seaplane of total mass m lands on a lake with initial speed of vi.

00:11 The only horizontal force on it is the resistance force on its pontoons from water.

00:17 The resistive force is proportional to the velocity of the sea plane.

00:21 And here is my equation.

00:23 Due to the second law applied to the plane is given in this equation.

00:35 There we go.

00:36 From the fundamental theorem of calculus, this differential equation implies that the speed change that i've given right here.

00:45 For part a, we're asked to carry out the integration to determine the speed of the c plane as a function of time.

00:53 Okay, so i am going to go ahead and write this down again right here.

01:09 Okay, the integral of dx over x is ln of x.

01:18 So let's do the ln of v.

01:25 We'll go to v and v -i equal negative b over m -t -t and zero.

01:49 Okay, so far so good.

01:52 Basic rules of logarithms.

02:20 And this will give me ln, v over v -i, is negative v -t over m.

02:28 And then take the natural log of each side for both sides, and we'll get v over v i equals and that'll be to the negative b t over m so b will equal v i e to the negative b t over m okay for part b we're asked to graph so this is my v this is my t this is v this is v i so this will go that's not very beautiful but it's good enough okay.

03:56 C.

04:00 Does c plane come to a complete stop after a finite interval of time? and we can see right here, we can see that no...

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