- #1

gelfand

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## Homework Statement

A submarine engine provides maximum constant force ##F## to propel it through the

water.

Assume that the magnitude of the resistive drag force of the water experienced

by the submarine is ##kv##, where ##k## is the drag coefficient and ##v## is the

instantaneous speed of the boat.

When the engine is switched on to full power, the submarine starts from rest and

moves horizontally in a straight line.

If the mass of the submarine is ##M##

1) Write the differential equation of motion of the submarine in terms of its

speed ##v##

2) Solve the equation of motion from part (1) to find the speed ##v(t)## as a

function of time ##t##

3) What is the maximum possible speed ##v_{max}## the submarine can reach with

this engine?

## Homework Equations

Newtons second states ##F = ma##. Then given force ##F## I have ##F = ma##.

## The Attempt at a Solution

For the drag force ##-kv## I can write this as ##-Kmv## where ##K## is some constant

such that ##-Kmv = -kv##.

This gives me

$$

ma = -Kmv

$$

Dividing by ##m## and noting that ##a = \frac{dv}{dt}## gives

$$

\frac{dv}{dt} = -Kv

$$

Which is a differential equation of motion in terms of the speed ##v##.

Solving this differential equation as

$$

\frac{1}{v} dv = -K dt

$$

Then integrate both sides for

$$

\ln(v) = -K t + C_1

$$

Here I can take the exponential of each side and note that ##e^{x + b} = Ae^x##

where ##A,b## are constants.

So I have

$$

v = Ae^{-Kt}

$$For the maximum speed I need to consider where the force of motion by the engine

is balanced by the force of the drag.

At this point I will have

$$

kv_{max} = ma

$$

Using the solution from part (2) we can sub for ##v## as

$$

k\left( A e^{-K t} \right) = ma

$$I'm not sure how to obtain the maximum speed here though?

The expression can be rewritten as

$$

\frac{kA}{ma} = e^{Kt}

$$

Given ##K## is some constant this is just (where ##D## is some constant)

$$

\frac{kA}{ma} = e^{t} e^{K} = De^{t}

$$

So

$$

\frac{kA}{Dma} =

e^{t}

$$

I don't think that this is correct, or even know if it makes sense.

It's suggesting that the constant ##k## multiplied by the constant ##A## divided by

the product of mass, acceleration and ##D## is equal to ##e^t##.

So I'm confused

Thanks

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