How to find r(t) by given:v(t) r(2) ?

[Lenjaku]

Homework Statement

Calculate the position vector at any given moment ( r(t) )

v(t)=(6t^2+4t)i+(3t-2)j+(12t^3-t^3)k m/s
r(t=2)= (3,-0.5,-2)

Homework Equations

r(t) = r0+ integral(v(t))

The Attempt at a Solution

I tried to integrate v(t) and get r(t) but I don't know how to calculate r0.
what am I supposed to do with t=2 I was given?
if I do integral of v(t) and calculate the it by t=2 - t=0, what would be the meaning of this?
t=0 doesn;t have to give r(0)=(0,0,0)

We learned to get the vectors with known starting points so I am puzzled about finding them myself and unfortunately all the questions deal with finding them -.-
I seek to learn the meaning of the whole picture.
then I can solve the rest.

 

[tiny-tim]

Welcome to PF!

Hi Lenjaku! Welcome to PF!

In the formula r(t) = r(0) + ∫ (v(t)) dt,

the limits of integration have to be 0 and t …

r(t) = r(0) + ∫₀^t (v(t)) dt​

But you can substitute any value a of t …

r(t) = r(a) + ∫_a^t (v(t)) dt ​

 

[Lenjaku]

Unfortunately I am still trying to figure out things we only had 1 lecture about this.
If I take the integral of t=2 and reduce t=0 it woudlnt give me anything (I think).

Just to see what happens I integrated v(t) and placed t=2. the numbers are nothing near r(2).

it turned out to be:
∫v(t)=24i+(-1)j+12k
r(2)=3i-0.5j-2k

What would be the meaning of this difference?

Both is the same time,but apparently the starting point affects the result...did I calculate ∫v(2) as r(0,0,0) starting point? if so why it turned to be so if at all?

 

[tiny-tim]

Hi Lenjaku!

Just to see what happens I integrated v(t) and placed t=2. the numbers are nothing near r(2).

it turned out to be:
∫v(t)=24i+(-1)j+12k
r(2)=3i-0.5j-2k

not following you

v(t)=(6t²+4t)i+(3t-2)j+(12t³-t³)k m/s

(hmm … that last bracket looks wrong )

can you show all the steps for the way you calculated ∫ v(t) dt?

 

[Lenjaku]

I integrated v(t) as:
(6t³)/3 + (4t²)/2 , (3t²)/2 - 2t,(12t⁴)/4 - (9t⁴)/4

And then placed t=2.

r(2) was given.

I simply wanted to see what it turns to be to understand what is going on there...

I still don't understand how to find r(t) at any given time when I am only given r(t=2) and no r(t=0) how do I find r(t=0) (do I need it?, if not why)?

I think I need it to put it in the equation:

r(t=0) + ∫v(t)
I have v(t) I can get its integral but what do I do with the given r(t=2)?
Why was I given this for?

 

[tiny-tim]

v(t₂) - v(t₁) = [(6t³)/3 + (4t²)/2 , (3t²)/2 - 2t,(12t⁴)/4 - (9t⁴)/4]_t1^t₂ …

you subtract the initial value from the final value

 

[Lenjaku]

I am sorry I don;t get you.
The only t I was given is t=2.
And I was give the components of r(t=2) so I assume the answer should do something with it.

I don;t know what r(t=0) is.

What do I do with the r(t=2) I was given?

Doing as u suggested was my initiate approach with no any further thinking it seems logical but it leaves me with too many questions.

I am thinking of x=x₀+v₀t
taking v as constant. So v₀=v.