Particle trajectories


by Bipolarity
Tags: particle, trajectories
Bipolarity
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#1
Nov16-12, 01:34 PM
P: 783
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

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

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
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Ray Vickson
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#2
Nov16-12, 03:30 PM
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Quote Quote by Bipolarity View Post
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

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

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

RGV
LCKurtz
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#3
Nov16-12, 03:34 PM
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Quote Quote by Bipolarity View Post
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

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

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
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.

Bipolarity
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#4
Nov16-12, 11:26 PM
P: 783

Particle trajectories


Quote Quote by LCKurtz View Post
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.
Where in my equation did I add vectors with scalars? Every term in my equation is a position vector...

BiP
haruspex
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#5
Nov17-12, 12:05 AM
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Quote Quote by Bipolarity View Post
the particle travels in a straight line, meaning its unit tangent vector is constant and that its motion is nonperiodic
If it travels in a straight line then its acceleration must always be collinear with the velocity.
LCKurtz
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Nov17-12, 04:22 PM
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Quote Quote by Bipolarity View Post

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


BiP
Quote Quote by LCKurtz View Post
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.
Quote Quote by Bipolarity View Post
Where in my equation did I add vectors with scalars? Every term in my equation is a position vector...

BiP
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.


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