# Problem Finding the Speed of Particle (Vector)

• Oct 27th 2012, 02:23 PM
Beevo
Problem Finding the Speed of Particle (Vector)
Hey everyone,
I am having a bit of difficulty solving this problem. It says:
A particle has position function $\ r(t) = (2\sqrt{2})i + (e^{2t})j + (e^{-2t})k$. What is its speed at time t?

Well I first took the derivative of the function. Then I took the magnitude to find the length which corresponds to the speed. I got my answer to be
$\sqrt{8+4e^{2t}+4e^{-2t}}$, is this answer correct? If so, how would I further simplify it? If it is incorrect, what am I doing wrong?

Any help and feedback is greatly appreciated, thanks.
• Oct 27th 2012, 03:32 PM
Soroban
Re: Problem Finding the Speed of Particle (Vector)
Hello, Beevo!

Quote:

A particle has position function: $\ r(t) \:=\: (2\sqrt{2})i + (e^{2t})j + (e^{-2t})k$.
What is its speed at time $t$?

Well, I first took the derivative of the function.
Then I took the magnitude to find the length which corresponds to the speed.
I got my answer to be: $\sqrt{8+4e^{2t}+4e^{-2t}}$.

Is this answer correct? . Yes!
If so, how would I further simplify it?

It can be simplified . . .

$\sqrt{4e^{2t} + 8 + 4e^{-2t}} \;=\;\sqrt{4(e^{2t} + 2 + e^{-2t})} \;=\;2\sqrt{e^{2t} + 2 + e^{-2t}}$

. . . . . . . . . . . . . $=\;2\sqrt{(e^t + e^{-t})^2} \;=\;2(e^t + e^{-t})$

• Oct 27th 2012, 03:34 PM
topsquark
Re: Problem Finding the Speed of Particle (Vector)
Quote:

Originally Posted by Beevo
Hey everyone,
I am having a bit of difficulty solving this problem. It says:
A particle has position function $\ r(t) = (2\sqrt{2})i + (e^{2t})j + (e^{-2t})k$. What is its speed at time t?

Well I first took the derivative of the function. Then I took the magnitude to find the length which corresponds to the speed. I got my answer to be
$\sqrt{8+4e^{2t}+4e^{-2t}}$, is this answer correct? If so, how would I further simplify it? If it is incorrect, what am I doing wrong?

Any help and feedback is greatly appreciated, thanks.

Looks good to me, unless you have to use hyperbolic functions to simplify.

-Dan

PS okay I should have seen that one coming Soroban! Thanks.
• Oct 27th 2012, 03:48 PM
Beevo
Re: Problem Finding the Speed of Particle (Vector)
Thanks for the feedback guys. The simplification part just kind of threw me off.
• Oct 27th 2012, 04:18 PM
HallsofIvy
Re: Problem Finding the Speed of Particle (Vector)
Okay, I'm confused! Where did that "8" come from? With the position $r(t)= 2\sqrt{2}\vec{i}+ e^{2t}\vec{j}+ e^{-2t}\vec{k}$,I get the velocity to be $2e^{2t}\vec{j}- 2e^{-2t}\vec{k}$ and so the speed is $\sqrt{4e^{4t}+ 4e^{-4t}}= 2\sqrt{e^{4t}+ e^{-4t}}$
• Oct 27th 2012, 04:23 PM
Beevo
Re: Problem Finding the Speed of Particle (Vector)
Quote:

Originally Posted by HallsofIvy
Okay, I'm confused! Where did that "8" come from? With the position $r(t)= 2\sqrt{2}\vec{i}+ e^{2t}\vec{j}+ e^{-2t}\vec{k}$,I get the velocity to be $2e^{2t}\vec{j}- 2e^{-2t}\vec{k}$ and so the speed is $\sqrt{4e^{4t}+ 4e^{-4t}}= 2\sqrt{e^{4t}+ e^{-4t}}$

My fault, there should be a 't' after $\2\sqrt{2}$