# Showing acceleration is constant

1. Jan 30, 2009

### Ravenatic20

I hope I posted in the right place. Sorry in advanced.

1. The problem statement, all variables and given/known data
A buzzing fly moves in a helical path given by the equation
r(t) = ib sin $$\omega$$t + jb cos $$\omega$$t + kct$$^{2}$$
Show that the magnitude of the acceleration of the fly is constant, provided b, $$\omega$$, and c are constant.

3. The attempt at a solution
x = b sin $$\omega$$t
y = b cos $$\omega$$t
z = ct$$^{2}$$

In class we did a similar problem, but in that problem we had to find the trajectory in space. I'm a little slow, but it's just not helping me with this one. Same with the textbook. I'll go to my teacher if I have to.

I'm not asking someone to do the problem, just get me started. Once that happens, I'll try to go over it here in case I have more questions. Thank you!

2. Jan 30, 2009

### Dick

The acceleration is the second derivative of r(t) with respect to t. What is that?

3. Jan 31, 2009

### Ravenatic20

The first derivative:
ib$$\omega$$ cos $$\omega$$t - jb$$\omega$$ sin $$\omega$$t + 2kct

Second derivative:
-ib$$\omega$$$$^{2}$$ sin $$\omega$$t - jb$$\omega$$$$^{2}$$ cos $$\omega$$t + 2kc

Is that right?

4. Jan 31, 2009

### Nabeshin

Now find the magnitude...

5. Jan 31, 2009

### gabbagabbahey

Sure; that;s correct ....But very ugly!:yuck: Try writing the entire equation inside the [ tex] or [ itex] tags instead:

$$\mathbf{a}(t)=-b\omega^2\sin(\omega t)\mathbf{i}-b\omega^2\cos(\omega t)\mathbf{j}+2c\mathbf{k}$$

(You can click on the above equation to see the code that generated it)

Now, as Nabeshin said, calculate the magnitude

6. Feb 2, 2009

### Ravenatic20

Thank you guys :)

Now what is the first step in calculating the magnitude? I am used to plugging in numbers to do that.

7. Feb 2, 2009

### gabbagabbahey

You know what the x,y, and z-components of a are, so square them, add the squares, and take the square root as per usual.

$$||\mathbf{a}||=\sqrt{a_x^2+a_y^2+a_y^2}$$

8. Feb 2, 2009

### Ravenatic20

So I take $$\mathbf{a}$$ (the second derivative above), and factor in x for the first part. So it would look like this:

$$\mathbf{a_x}(t)=[-b\omega^2\sin(\omega t)\mathbf{i}-b\omega^2\cos(\omega t)\mathbf{j}+2c\mathbf{k}*\mathbf{b}sin(\omega t)]^2$$, where $$\mathbf{b}sin(\omega t)$$ is x.

Then do the same thing for y and z, add the terms up, and take the square root. Am I on the right track? Then I simplify as much as possible?

9. Feb 2, 2009

### gabbagabbahey

Huh?!

No! $a_x$ is the x-component of a....that's just $-b\omega^2\sin(\omega t)$....what are
$a_y$ and $a_z$?

10. Feb 2, 2009

### Ravenatic20

Thank you. I always over complicate things. I believe I know how to do it now.

To answer your question, y is $$-b\omega^2\cos(\omega t)$$ and z is $$2c$$
Now I square them, add them up, and take the square root of that sum (this:$$||\mathbf{a}||=\sqrt{a_x^2+a_y^2+a_y^2}$$). Correct?

11. Feb 2, 2009

### D H

Staff Emeritus
Correct. So what is the result?