find the velocity vector for an object

find the velocity vector for an object having

$\displaystyle \vec a(t) = e^t \hat{j} -32 \hat{k} $

if

$\displaystyle v(0) = 3 \hat{i} - 2 \hat{j} + \hat{k}$

Im not sure how to proceed. Would i integrate? How to handle the v(0) has me confused!

$\displaystyle v(t) = \int e^t \hat{j} -32 \hat{k}\ dt$

$\displaystyle v(t) = e^t j -32t k + c$

Im not sure if i am doing this correctly?

Re: find the velocity vector for an object

Quote:

Originally Posted by

**icelated** Im not sure if i am doing this correctly?

Yes. The initial velocity, $\displaystyle \mathbf v(0),$ is an *initial condition*. Use it to determine the constant of integration.

$\displaystyle \mathbf v(t) = e^t\mathbf j-32t\mathbf k + \mathbf C$

So

$\displaystyle \mathbf v(0) = 3\mathbf i - 2\mathbf j + \mathbf k$

$\displaystyle \Rightarrow e^0\mathbf j-32\cdot0\mathbf k + \mathbf C = 3\mathbf i - 2\mathbf j + \mathbf k$

$\displaystyle \Rightarrow\mathbf C = 3\mathbf i - 3\mathbf j + \mathbf k$

Re: find the velocity vector for an object

Would i then integrate again?

Here is the final answer, but i cant seem to get to it

$\displaystyle v(t) = 3i +(e^t - 3)j +( -32t +1)k$

Re: find the velocity vector for an object

Quote:

Originally Posted by

**icelated** Would i then integrate again?

Why would you?

Quote:

Originally Posted by

**icelated** Here is the final answer, but i cant seem to get to it

$\displaystyle v(t) = 3i +(e^t - 3)j +( -32t +1)k$

Just substitute what you (I) found for $\displaystyle \mathbf C$ back into the equation for $\displaystyle \mathbf v(t).$ That's it.

Re: find the velocity vector for an object

I can see that now! Thank you..

Problems like this in the book integrate. ybe if v(0) = 0?