Integrating Velocity When in Unit Vector Notation

[ThomasMagnus]

Homework Statement

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?

Homework Statement

N/A

Homework Equations

N/A

 

[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 traveling in the j component separately.

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

 

[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!

 

[Maybe_Memorie]

Yes, you can.