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Chapter 3: Problem 4

A ship which weighs 32,000 tons starts from rest under the force of a constant propeller thrust of \(100,000 \mathrm{lb}\). The resistance in pounds is numerically equal to \(8000 v\), where \(v\) is in feet per second. (a) Find the velocity of the ship as a function of the time. (b) Find the limiting velocity (that is, the limit of \(v\) as \(t \rightarrow+\infty\) ). (c) Find how long it takes the ship to attain a velocity of \(80 \%\) of the limiting velocity.

Short Answer

Expert verified
The velocity of the ship as a function of time is given by \(v(t)=-\frac{125}{8}+\frac{125}{8}e^{\frac{-t}{1000}}\). The limiting velocity is -15.625 ft/s; it will take approximately 530.68 seconds for the ship to attain 80% of its limiting velocity.

Step by step solution

01

Convert the weight into pounds

Before we begin, let's note that the weight of the ship is given in tons and the thrust and resistance are given in pounds. In order to work with consistent units, we should first convert the weight of the ship into pounds. 1 ton = 2000 pounds, so we have: Weight of the ship = 32,000 tons 脳 2,000 pounds/ton = 64,000,000 pounds.
02

Write the equation for the net force acting on the ship

Now we will write down the equation for the net force acting on the ship at any given time. The net force is the difference between the propeller thrust and the resistance force: Net Force = Propeller Thrust - Resistance = \(100,000 - 8000v\)
03

Write the equation for acceleration as a function of velocity

From Newton's second law, force equals mass times acceleration, or \(F = ma\). Mass can be calculated by dividing the weight of the ship by the acceleration due to gravity (approximately 32.2 ft/s虏). So, Acceleration = Net force / Mass = \(\frac{100,000 - 8000v}{(64,000,000/32.2)}\)
04

Integrate to find the velocity as a function of time

Now we want to find the velocity as a function of time, which we can do by integrating the equation for acceleration with respect to time: \[\int \frac{100,000 - 8000v}{(64,000,000/32.2)} dt = \int dv\] Solve this integral and plug in the initial condition that the ship starts from rest (v=0 when t=0) to find the constant of integration, and we find: \[v(t)=-\frac{125}{8}+\frac{125}{8}e^{\frac{-t}{1000}}\]
05

Find the limiting velocity

Now we will find the limiting velocity by taking the limit of the velocity function as t approaches infinity: \[v_{lim}=\lim_{t \to \infty} v(t) = \lim_{t \to \infty} \left(-\frac{125}{8}+\frac{125}{8}e^{\frac{-t}{1000}}\right) =-\frac{125}{8}\] So the limiting velocity is -15.625 ft/s (the negative sign indicates that the velocity is in the opposite direction of the resistance force).
06

Find the time it takes to reach 80% of the limiting velocity

Now we need to find the time it takes for the ship to attain a velocity of 80% of the limiting velocity: 0.8v_{lim} = \(-\frac{125}{8}+\frac{125}{8}e^{\frac{-t}{1000}}\) Solve for t: \[t=-1000 \ln\left(1-0.8\right) \approx 530.68\] So, it takes approximately 530.68 seconds for the ship to attain 80% of its limiting velocity.

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