# Particle trajectories

1. ### Bipolarity

794
1. The problem statement, all variables and given/known data

At time t = 0, a particle is located at the point (1, 2, 3). It travels
in a straight line to the point (4, 1,4), has speed 2 at (1, 2, 3) and
constant acceleration 3i - j + k. Find an equation for the position
vector r(t) of the particle at time t.

2. Relevant equations

3. The attempt at a solution
If we integrate the acceleration vector twice, we get the position vector (along with some constants we must determine). Plugging in the initial position vector gives us

$$r(t) = <1.5t^{2},-.5t^{2},.5t^{2}> + v_{0}t + <1,2,3>$$

The problem is I don't know how to find the initial velocity. That's all I need to complete the problem. I know the speed at time t=0, but not the velocity. I also know that the particle travels in a straight line, meaning its unit tangent vector is constant and that its motion is nonperiodic, but how does that help?

BiP

2. ### Ray Vickson

6,046
If a particle travels in a straight line, how are the velocity and acceleration vectors related?

RGV

3. ### LCKurtz

8,338
That equation doesn't make any sense because you can't add vectors and scalars. And remember you can always express a velocity vector as the speed times a unit vector in the correct direction.

4. ### Bipolarity

794
Where in my equation did I add vectors with scalars? Every term in my equation is a position vector...

BiP

5. ### haruspex

13,139
If it travels in a straight line then its acceleration must always be collinear with the velocity.

6. ### LCKurtz

8,338
Well, you mentioned that the initial speed was 2. Since there was no definition given for ##v_0## and it looks like a scalar, I assumed it was ##2##. And there is no notation to distinguish vectors from scalars, I assumed ##v_0 t## was a scalar.