Velocity function of a variable force

[Piamedes]

Homework Statement

A particle with a mass of 3kg is at rest at x = 3m, and then a force = (12 N/m)x is applied to it. What is the acceleration of the particle when it reaches x = 5m? Determine the v(t) for the particle.

Homework Equations

<br /> \sum F = ma<br />

The Attempt at a Solution

The first part was simple. I just solved for acceleration and got the same answer as the book, 20 m/s^2. However there was no answer for the second part so I want to know if I solved for v(t) properly.

<br /> a = (4 \frac {1}{s^2}) x<br />

<br /> a = \frac {d^2x} {dt^2}<br />

<br /> \frac {d^2x} {dt^2} - (4 \frac {1}{s^2}) x = 0<br />

Which is just a 2nd order differential equation, whose general solution is:
<br /> x = C_1 e^(\lambda_1 t) + C_2 e^(\lambda_2 t)<br />

Where lambda is the solution to the auxiliary equation,
<br /> \lambda^2 - 4\lambda = 0<br />

Therefore
<br /> x = C_1 e^(2 t) + C_2 e^(-2 t)<br />

and v(t) is
<br /> v = 2 C_1 e^(2 t) - 2 C_2 e^(-2 t)<br />

The problem stipulates that at x=3m, v=0, so when if I set t=0 at x=3m, then I can solve for C1 and C2
<br /> 3 = C_1 + C_2<br />

<br /> 0 = 2 C_1 - 2 C_2<br />

<br /> C_1 = \frac {3}{2} and C_2 = \frac {-3}{2}<br />

So v(t) becomes
<br /> v = (3 \frac {m}{s}) ( e^(2 t) + e^(-2 t) )<br />

Then I wanted to make it look nicer so I rewrote it as:
<br /> v = (6 \frac {m}{s}) \cosh {(2 \frac {1}{s} t)}<br />

Is this the proper solution to the question?
If not could someone please explain to me how to get the correct one. Thank you for the help.

 

[alphysicist]

Hi Piamedes,

Homework Statement

A particle with a mass of 3kg is at rest at x = 3m, and then a force = (12 N/m)x is applied to it. What is the acceleration of the particle when it reaches x = 5m? Determine the v(t) for the particle.

Homework Equations

<br /> \sum F = ma<br />

The Attempt at a Solution

The first part was simple. I just solved for acceleration and got the same answer as the book, 20 m/s^2. However there was no answer for the second part so I want to know if I solved for v(t) properly.

<br /> a = (4 \frac {1}{s^2}) x<br />

<br /> a = \frac {d^2x} {dt^2}<br />

<br /> \frac {d^2x} {dt^2} - (4 \frac {1}{s^2}) x = 0<br />

Since the problem involves v and x, I would write

<br /> \frac {d^2x} {dt^2} <br />

in a form that does not have t in it:

<br /> \frac {d^2x} {dt^2} =v \frac{dv}{dx}<br />

Then you can integrate over v and x directly. (But you might also want to drop the (1/s^2) that is inside your equation.)

 

[Piamedes]

I tried doing that at first, but since the problem required v as a function of time when I integrated again to solve for x as a function of t it came out to
<br /> \ln [x + \sqrt{x^2 - 9}] = 2 t<br />

So when I tried solving for x it I lost the +/- portion of the square. I went back and used the general solution because it seemed easier than trying to figure out a mistake in my page of algebra work.

Does doing it different ways yield different answers?

 

[alphysicist]

I tried doing that at first, but since the problem required v as a function of time when I integrated again to solve for x as a function of t it came out to
<br /> \ln [x + \sqrt{x^2 - 9}] = 2 t<br />

So when I tried solving for x it I lost the +/- portion of the square. I went back and used the general solution because it seemed easier than trying to figure out a mistake in my page of algebra work.

Does doing it different ways yield different answers?

I'm not sure I understand what happened, but different way should definitely give the same answer.

Looking over your original post, I think your values for C1 and C2 are incorrect. Your second equation for them is:

<br /> 0=2 C_1 - 2 C_2<br />
so they must be equal.

 

[Piamedes]

oops. I solved it all the way through to the end and messed up some algebra.
Its should be
<br /> C_1 = C_2 = \frac {3}{2}<br />

Which means that
<br /> x = (\frac {3}{2} \frac {m}{s}) (e^(2 t) + e^(-2 t) ) = (\frac {3}{2} \frac {m}{s}) \cosh {(2t)}<br />

<br /> x = (3 \frac{m}{s}) (e^(2 t) - e^(-2 t) ) = (3 \frac {m}{s}) \sinh {(2t)}<br />

<br /> a = (6 \frac {m}{s}) (e^(2 t) + e^(-2 t) ) = (6 \frac {m}{s}) \cosh {(2t)}<br />

And if I check from the original equation
<br /> a = 4x<br />

<br /> a = 4 ( (\frac {3}{2} \frac {m}{s}) \cosh {(2t)} ) = 4x<br />

By showing that my solution fits the given parameters, is that proof that the answer is correct or am I still missing something.
Thanks for all the help so far.

 

[alphysicist]

oops. I solved it all the way through to the end and messed up some algebra.
Its should be
<br /> C_1 = C_2 = \frac {3}{2}<br />

Which means that
<br /> x = (\frac {3}{2} \frac {m}{s}) (e^(2 t) + e^(-2 t) ) = (\frac {3}{2} \frac {m}{s}) \cosh {(2t)}<br />

You have a small error here. Notice that the exponential form gives x=3 at t=0, while the cosh from doesn't.

 

[Piamedes]

Thanks for catching that error. I must have forgotten to factor out the 1/2 when I transformed it into cosh.

It should be
<br /> x = (3 \frac {m}{s}) \cosh {(2t)}<br />

Thanks for the help.

 

[alphysicist]

Sure, glad to help! (And the way you put all the details of your work in your posts definitely made it easy to read through.)