/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 37 Sinking Boat As a boat weighing ... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 37

Sinking Boat As a boat weighing \(1000 \mathrm{lb}\) sinks in water from rest, it is acted upon by a buoyant force of \(200 \mathrm{lb}\) and a force of water resistance in pounds that is numerically equal to \(100 v\), where \(v\) is in feet per second. Find the distance traveled by the boat after \(4 \mathrm{sec}\). What is its limiting velocity?

Short Answer

Expert verified
The distance traveled by the boat after 4 seconds is 2560 feet, and its limiting velocity is 8 feet per second.

Step by step solution

01

Identify the forces acting on the boat

The boat is acted upon by three forces: its weight (W), the buoyant force (B), and the water resistance (R). The weight is given as 1000 lb, the buoyant force is 200 lb, and the water resistance is numerically equal to 100v, where v is the velocity in feet per second. So, we can write the forces as: Weight (W) = 1000 lb Buoyant Force (B) = 200 lb Water Resistance (R) = 100v lb
02

Calculate the net force acting on the boat

Since the boat is sinking in the water, the net force (F_net) acting on it can be represented as the difference between its weight and the sum of buoyant force and water resistance. Therefore, F_net can be given as: F_net = W - (B + R) Substitute the values of W, B, and R: F_net = 1000 - (200 + 100v) Simplify the equation: F_net = 800 - 100v
03

Apply Newton's second law to determine the acceleration

Newton's second law states that F_net = ma, where m is the mass of the boat and a is its acceleration. We are given W = 1000 lb, and since the acceleration due to gravity (g) is 32 ft/s^2, we can find the mass (m) as follows: m = W / g m = 1000 / 32 m = 31.25 slugs (mass unit in the English system) Now, substitute the value of F_net and m in the equation F_net = ma to find the acceleration (a): 31.25a = 800 - 100v Divide both sides by 31.25: a = (800 - 100v) / 31.25
04

Integrate the acceleration with respect to time to find the velocity

To find the velocity (v) as a function of time (t), we need to integrate the acceleration (a) with respect to time: v(t) = 鈭�(800 - 100v) / 31.25 dt Let u = 100v, then du/dt = 100 and dv = du/100. Substituting, we can write: v(t) = (1/100) 鈭�(800 - u) / 31.25 du Integrate: v(t) = (1/100)(-u^2/62.5 + 800u) + C To solve for the constant C, we use the initial condition that the boat is sinking from rest, which means v(0) = 0: 0 = (1/100)(-0 + 800*0) + C C = 0 So the final velocity equation is: v(t) = (1/100)(-u^2/62.5 + 800u) Now, substitute u = 100v back into the equation: v(t) = (1/100)(-100^2v^2/62.5 + 80000v) Simplify: v(t) = -v^3 + 800v
05

Integrate the velocity with respect to time to find the distance

To find the distance (d) traveled by the boat as a function of time (t), we need to integrate the velocity (v) with respect to time: d(t) = 鈭�(-v^3 + 800v) dt Integrate: d(t) = -1/4v^4 + 400v^2 + D To solve for the constant D, we use the initial condition that the boat starts from rest, so the distance traveled at the beginning is zero: d(0) = 0: 0 = -1/4(0)^4 + 400(0)^2 + D D = 0 So the final distance equation is: d(t) = -1/4v^4 + 400v^2 Now, we need to find the distance traveled by the boat after 4 seconds. Substitute the given time (t = 4 seconds) in the distance equation: d(4) = -1/4(4)^4 + 400(4)^2 Calculate the result: d(4) = 2560 feet
06

Find the limiting velocity

The limiting velocity (v_limit) is the maximum velocity the boat can attain when all the forces acting on it are balanced. To find the limiting velocity, we set the acceleration (a) to zero: 0 = (800 - 100v_limit) / 31.25 Multiply both sides by 31.25: 0 = 800 - 100v_limit Solve for v_limit: 100v_limit = 800 v_limit = 800 / 100 v_limit = 8 ft/s The limiting velocity of the sinking boat is 8 feet per second. In conclusion, the distance traveled by the boat after 4 seconds is 2560 feet, and its limiting velocity is 8 feet per second.

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Key Concepts

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

Buoyant Force
In physics, the concept of buoyant force is crucial when studying objects submerged in fluid, such as water. A buoyant force is the upward force exerted by a fluid to oppose the weight of an object placed in it. This force arises because of the pressure differences within the fluid surrounding the object.
When a boat is submerged, part of the water is displaced. According to Archimedes' Principle, the buoyant force is equal to the weight of the fluid displaced by the object. For the sinking boat scenario, the buoyant force is given as 200 lb. This upward force works against the pull of gravity and reduces the net force acting on the boat.
This reduction affects how quickly and how far the boat will sink. Without this upward force, the boat would descend much faster. The buoyant force is equivalent to the weight of the displaced water, offering a counteractive force that partly supports the boat's weight.
Limiting Velocity
The concept of limiting velocity refers to the maximum speed an object will reach, due to counteracting forces balancing out the force of gravity. In this context, when the boat sinks, it is subjected to several forces:
  • Its own weight pulling it downwards
  • The buoyant force pushing it upwards
  • The water resistance acting against its motion
Initially, the boat accelerates downward until the upward forces, buoyant force and water resistance, balance the downward gravitational force. At this point, the net force becomes zero, meaning there is no further acceleration, and the boat achieves a constant velocity 鈥� the limiting velocity.
For the sinking boat, the limiting velocity was calculated to be 8 feet per second. This is the maximum speed the boat will attain as it continues to sink, and it remains constant as long as the external conditions remain the same. Understanding limiting velocity is important as it helps predict the long-term motion behavior of objects in fluid environments.
Water Resistance
Water resistance plays a significant role in affecting the motion of objects moving through water. It acts as a drag force that opposes the motion of the object through the water. In the exercise, the water resistance is given as proportional to the velocity of the boat, represented by the equation \(R = 100v\), where \(v\) is the velocity.
This resistance is due to the water molecules colliding and interacting with the surface of the sinking boat, gradually slowing its descent. The faster the boat moves, the greater is the force of water resistance. Hence, the resistance increases with an increase in speed, acting as a dynamic force that tries to bring the object within the fluid to rest.
Water resistance is a form of friction and can significantly affect an object's speed, especially at higher velocities. It is a key factor when determining the limiting velocity because it will continue to balance against the net downward force at higher speeds, preventing further acceleration of the sinking boat.

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