Integrating Velocity When in Unit Vector Notation

1. Feb 5, 2014

ThomasMagnus

1. The problem statement, all variables and given/known data

Say for example, a particles velocity was given by the following equation:

$\vec{V}$(t) = (2t2-4t3)$\hat{i}$ - (6t +3)$\hat{j}$ + 6$\hat{k}$

If I wanted to find the displacement of the particle between t=1s and t=3s, could I just integrate like this?

$\int \vec{V}$= (2t3/3 - t^4)$\hat{i}$ - (3t2 +3t)$\hat{j}$ + 6t $\hat{k}$ evaluated between 1.00 and 3.00

= (-63i)-36j + 18k)-(2/3-1)i+(6j)-6k= -63.3i - 30j + 12k.

Is this the correct way to do this?

1. The problem statement, all variables and given/known data

N/A

2. Relevant equations

N/A

Last edited: Feb 5, 2014
2. Feb 5, 2014

Maybe_Memorie

Yep, that's correct.

As for why it's correct, suppose the particle's velocity was just 6i, so the distance is only changing in the i direction so you only integrate in that direction. Then if it's velocity was 6i + 3j, the total displacement is the same as moving the i component, then travelling in the j component separately.

The total displacement is just the vector sum, hence why your integration is correct.

3. Feb 5, 2014

ThomasMagnus

So can you just treat each unit vector separately and integrate and evaluate each individually, then combine them all to find the displacement vector?

Thanks!

4. Feb 5, 2014

Maybe_Memorie

Yes, you can.