# Thread: Position, velocity and acceleration of a particle

1. ## Position, velocity and acceleration of a particle

I have a problem for uni that I don't really know how to approach. The problem is:

"The velocity of a particle at time t = v(t) = sinti + costj - 10tk. What is the particle's acceleration, a(t) = dv/dt as a function of time? Given that at t = 0 the particle is at (1,0,0), find its position at time t."

I understand the relationship between position, velocity and acceleration. i.e. - If x(t) is the position at time t, then the velocity v(t) = dx/dt and the acceleration a(t) = dv/dt. What I don't understand is how to treat the i, j and k in the expression? Just for my own sanity I'm thinking of t like I'd usually think about an x variable, but are i, j and k supposed to be treated as constants? If they didn't exist, I'd have no problem integrating/differentiatiating, but their occurence has completely thrown me off. Any help would be greatly appreciated.

2. Originally Posted by drew.walker
I have a problem for uni that I don't really know how to approach. The problem is:

"The velocity of a particle at time t = v(t) = sinti + costj - 10tk. What is the particle's acceleration, a(t) = dv/dt as a function of time? Given that at t = 0 the particle is at (1,0,0), find its position at time t."

I understand the relationship between position, velocity and acceleration. i.e. - If x(t) is the position at time t, then the velocity v(t) = dx/dt and the acceleration a(t) = dv/dt. What I don't understand is how to treat the i, j and k in the expression? Just for my own sanity I'm thinking of t like I'd usually think about an x variable, but are i, j and k supposed to be treated as constants? If they didn't exist, I'd have no problem integrating/differentiatiating, but their occurence has completely thrown me off. Any help would be greatly appreciated.
If $r = x_1(t) i + x_2(t) j + x_3(t) k$ then $\frac{dr}{dt} = \frac{d x_1}{dt} i + \frac{d x_2}{dt} j + \frac{dx_3}{dt} k$ .

3. Your response is a little over my head, but would I be right in saying $\frac {dv}{dt} = costi - sintj - 10k$

4. Originally Posted by drew.walker
Your response is a little over my head, but would I be right in saying $\frac {dv}{dt} = costi - sintj - 10k$
Yes.

5. Thanks for the help. My only remaining confusion lies with the second part of the question in relation to the position at time t.

I think I'm right in saying that $x(t) = \int v(t) dt$, which I thought would mean that $x(t) = -costi + sintj - 5t^2k$

However, -cos(0) = -1 (which contradicts the position at t = 0 of (1,0,0)). Is there something wrong with the logic I've applied here? Unfortunately I'm mainly doing this based on intuition as our course material is pretty poor.

6. Originally Posted by drew.walker
Thanks for the help. My only remaining confusion lies with the second part of the question in relation to the position at time t.

I think I'm right in saying that $x(t) = \int v(t) dt$, which I thought would mean that $x(t) = -costi + sintj - 5t^2k$

However, -cos(0) = -1 (which contradicts the position at t = 0 of (1,0,0)). Is there something wrong with the logic I've applied here? Unfortunately I'm mainly doing this based on intuition as our course material is pretty poor.
$\vec{r} = -\cos (t) \vec{i} + \sin (t) \vec{j} - 5 t^2 \vec{k} + \vec{C}$ where $\vec{C}$ is a constant vector.

Given: At t = 0, $\vec{r} = \vec{i}$.

But at t = 0, $\vec{r} = -\vec{i} + \vec{C}$.

Therefore $\vec{i} = -\vec{i} + \vec{C} \Rightarrow \vec{C} = 2 \vec{i}$.

Therefore $\vec{r} = -\cos (t) \vec{i} + \sin (t) \vec{j} - 5 t^2 \vec{k} + 2 \vec{i} = [2 -\cos (t)] \vec{i} + \sin (t) \vec{j} - 5 t^2 \vec{k}$.